PermPal Database

Av(12354, 12453, 13254, 13452, 14253, 14352, 23154, 24153)

Counting Sequence

1, 1, 2, 6, 24, 112, 568, 3030, 16732, 94792, 547836, 3217248, 19144104, 115177418, 699451912, ...

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Implicit Equation for the Generating Function

\(\displaystyle \left(3 x -2\right) F \left(x \right)^{3}+\left(2 x^{2}-11 x +7\right) F \left(x \right)^{2}+\left(3 x^{3}-6 x^{2}+13 x -8\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}-2 x +3\right) = 0\)

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Recurrence

\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 568\)
\(\displaystyle a(7) = 3030\)
\(\displaystyle a(8) = 16732\)
\(\displaystyle a(9) = 94792\)
\(\displaystyle a(10) = 547836\)
\(\displaystyle a(11) = 3217248\)
\(\displaystyle a(12) = 19144104\)
\(\displaystyle a(13) = 115177418\)
\(\displaystyle a(14) = 699451912\)
\(\displaystyle a{\left(n + 15 \right)} = - \frac{4212 n \left(n - 1\right) a{\left(n \right)}}{\left(n + 14\right) \left(n + 15\right)} + \frac{9 n \left(3694 n + 3215\right) a{\left(n + 1 \right)}}{\left(n + 14\right) \left(n + 15\right)} + \frac{33 \left(n + 13\right) a{\left(n + 14 \right)}}{2 \left(n + 15\right)} + \frac{\left(55 n^{2} - 319 n - 11058\right) a{\left(n + 12 \right)}}{2 \left(n + 14\right) \left(n + 15\right)} - \frac{\left(179 n^{2} + 4372 n + 26628\right) a{\left(n + 13 \right)}}{2 \left(n + 14\right) \left(n + 15\right)} + \frac{\left(4519 n^{2} + 105374 n + 607776\right) a{\left(n + 11 \right)}}{2 \left(n + 14\right) \left(n + 15\right)} - \frac{3 \left(5162 n^{2} + 104351 n + 524851\right) a{\left(n + 10 \right)}}{\left(n + 14\right) \left(n + 15\right)} - \frac{6 \left(21802 n^{2} + 59060 n + 37047\right) a{\left(n + 2 \right)}}{\left(n + 14\right) \left(n + 15\right)} + \frac{12 \left(27020 n^{2} + 123649 n + 138730\right) a{\left(n + 3 \right)}}{\left(n + 14\right) \left(n + 15\right)} + \frac{\left(123739 n^{2} + 2176772 n + 9551028\right) a{\left(n + 9 \right)}}{2 \left(n + 14\right) \left(n + 15\right)} - \frac{\left(174932 n^{2} + 2645650 n + 9992997\right) a{\left(n + 8 \right)}}{\left(n + 14\right) \left(n + 15\right)} - \frac{3 \left(365173 n^{2} + 2375291 n + 3833404\right) a{\left(n + 4 \right)}}{2 \left(n + 14\right) \left(n + 15\right)} + \frac{\left(656008 n^{2} + 5581025 n + 11829648\right) a{\left(n + 5 \right)}}{\left(n + 14\right) \left(n + 15\right)} + \frac{\left(732431 n^{2} + 9372340 n + 29959236\right) a{\left(n + 7 \right)}}{2 \left(n + 14\right) \left(n + 15\right)} - \frac{\left(1141189 n^{2} + 12094445 n + 31995264\right) a{\left(n + 6 \right)}}{2 \left(n + 14\right) \left(n + 15\right)}, \quad n \geq 15\)

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This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 140 rules.

Finding the specification took 38976 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{42}\! \left(x , y\right) F_{79}\! \left(x \right)\\ F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{54}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{53}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{55}\! \left(x \right) &= x\\ F_{56}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= -\frac{-F_{61}\! \left(x , y\right) y +F_{61}\! \left(x , 1\right)}{-1+y}\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x , y\right)\\ F_{63}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{55}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{61}\! \left(x , 1\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{64}\! \left(x \right)\\ F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right) F_{79}\! \left(x \right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{55}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{55}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{55}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{74}\! \left(x \right) F_{88}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{88}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= y F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\ F_{95}\! \left(x \right) &= 0\\ F_{96}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{55}\! \left(x \right)+F_{94}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{117}\! \left(x , y\right)\\ F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{108}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= F_{107}\! \left(x \right) F_{12}\! \left(x , y\right)\\ F_{107}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{112}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{111}\! \left(x , y\right) &= F_{110}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{111}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\ F_{115}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{133}\! \left(x , y\right)\\ F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{123}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{132}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= F_{123}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\ F_{126}\! \left(x , y\right) &= -\frac{y \left(F_{10}\! \left(x , 1\right)-F_{10}\! \left(x , y\right)\right)}{-1+y}\\ F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right) F_{130}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{129}\! \left(x , y\right) &= y x\\ F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)\\ F_{131}\! \left(x , y\right) &= -\frac{-y F_{39}\! \left(x , y\right)+F_{39}\! \left(x , 1\right)}{-1+y}\\ F_{132}\! \left(x , y\right) &= F_{131}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{133}\! \left(x , y\right) &= F_{131}\! \left(x , y\right) F_{97}\! \left(x , y\right)\\ F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right)\\ F_{135}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{112}\! \left(x \right) F_{138}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{138}\! \left(x \right) &= \frac{F_{139}\! \left(x \right)}{F_{55}\! \left(x \right)}\\ F_{139}\! \left(x \right) &= F_{2}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 150 rules.

Finding the specification took 99765 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{145}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= y x\\ F_{27}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= x\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{31}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{125}\! \left(x , y\right)+F_{141}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{44}\! \left(x , y\right)+F_{48}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{43}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{33}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{53}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{58}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= -\frac{y \left(F_{33}\! \left(x , 1\right)-F_{33}\! \left(x , y\right)\right)}{-1+y}\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{64}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{70}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{63}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= -\frac{y \left(F_{18}\! \left(x , 1\right)-F_{18}\! \left(x , y\right)\right)}{-1+y}\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{77}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{79}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{77}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{81}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{63}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= y F_{89}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{24}\! \left(x \right)+F_{95}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{98}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= y x\\ F_{99}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{96}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= F_{93}\! \left(x , y\right) F_{98}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{111}\! \left(x , y\right)\\ F_{104}\! \left(x , y\right) &= -\frac{-y F_{103}\! \left(x , y\right)+F_{103}\! \left(x , 1\right)}{-1+y}\\ F_{105}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)+F_{108}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\ F_{107}\! \left(x , y\right) &= -\frac{F_{33}\! \left(x , 1\right)-F_{33}\! \left(x , y\right)}{-1+y}\\ F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{110}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= -\frac{-F_{16}\! \left(x , y\right) y +F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{31}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\ F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{114}\! \left(x , y\right)+F_{118}\! \left(x , y\right)+F_{24}\! \left(x \right)\\ F_{113}\! \left(x , y\right) &= F_{112}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{115}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{116}\! \left(x , y\right)+F_{117}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{115}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{117}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{118}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{28}\! \left(x \right)\\ F_{122}\! \left(x , y\right) &= F_{123}\! \left(x , y\right)+F_{124}\! \left(x , y\right)+F_{24}\! \left(x \right)\\ F_{123}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{81}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{126}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{139}\! \left(x , y\right)\\ F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{133}\! \left(x , y\right)\\ F_{131}\! \left(x , y\right) &= F_{132}\! \left(x \right) F_{46}\! \left(x , y\right)\\ F_{132}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\ F_{133}\! \left(x , y\right) &= F_{134}\! \left(x , y\right)\\ F_{134}\! \left(x , y\right) &= F_{132}\! \left(x \right) F_{135}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{136}\! \left(x , y\right) &= F_{135}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)\\ F_{137}\! \left(x , y\right) &= F_{138}\! \left(x \right)+F_{55}\! \left(x , y\right)\\ F_{138}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{139}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)\\ F_{140}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{141}\! \left(x , y\right) &= F_{142}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{142}\! \left(x , y\right) &= -\frac{-y F_{34}\! \left(x , y\right)+F_{34}\! \left(x , 1\right)}{-1+y}\\ F_{143}\! \left(x , y\right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\ F_{145}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x , 1\right)\\ F_{147}\! \left(x , y\right) &= y F_{103}\! \left(x , y\right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{2}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 142 rules.

Finding the specification took 38976 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{42}\! \left(x , y\right) F_{79}\! \left(x \right)\\ F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{57}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{54}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{53}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{55}\! \left(x \right) &= x\\ F_{56}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= -\frac{-F_{61}\! \left(x , y\right) y +F_{61}\! \left(x , 1\right)}{-1+y}\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x , y\right)\\ F_{63}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{55}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{61}\! \left(x , 1\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{64}\! \left(x \right)\\ F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right) F_{79}\! \left(x \right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{55}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{55}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{55}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{74}\! \left(x \right) F_{88}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{88}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{94}\! \left(x \right) &= 0\\ F_{95}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{98}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)+F_{106}\! \left(x , y\right)\\ F_{104}\! \left(x , y\right) &= F_{105}\! \left(x \right) F_{12}\! \left(x , y\right)\\ F_{105}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\ F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{110}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{109}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{109}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\ F_{113}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{136}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right)+F_{131}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right) F_{30}\! \left(x , y\right)\\ F_{121}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)+F_{130}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{123}\! \left(x , y\right)\\ F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)+F_{125}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= -\frac{y \left(F_{10}\! \left(x , 1\right)-F_{10}\! \left(x , y\right)\right)}{-1+y}\\ F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)\\ F_{126}\! \left(x , y\right) &= F_{127}\! \left(x , y\right) F_{128}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{127}\! \left(x , y\right) &= y x\\ F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)\\ F_{129}\! \left(x , y\right) &= -\frac{-y F_{39}\! \left(x , y\right)+F_{39}\! \left(x , 1\right)}{-1+y}\\ F_{130}\! \left(x , y\right) &= F_{129}\! \left(x , y\right) F_{55}\! \left(x \right)\\ F_{131}\! \left(x , y\right) &= F_{129}\! \left(x , y\right) F_{132}\! \left(x , y\right)\\ F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)+F_{55}\! \left(x \right)\\ F_{133}\! \left(x , y\right) &= F_{134}\! \left(x , y\right)+F_{135}\! \left(x , y\right)+F_{94}\! \left(x \right)\\ F_{134}\! \left(x , y\right) &= F_{132}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{135}\! \left(x , y\right) &= F_{55}\! \left(x \right) F_{97}\! \left(x , y\right)\\ F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)\\ F_{137}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{110}\! \left(x \right) F_{140}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{140}\! \left(x \right) &= \frac{F_{141}\! \left(x \right)}{F_{55}\! \left(x \right)}\\ F_{141}\! \left(x \right) &= F_{2}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 78 rules.

Finding the specification took 29701 seconds.

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This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 148 rules.

Finding the specification took 99765 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{146}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{143}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= y x\\ F_{27}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= x\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{31}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{4}\! \left(x \right)\\ F_{34}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{123}\! \left(x , y\right)+F_{139}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{44}\! \left(x , y\right)+F_{48}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{43}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{33}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{53}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{58}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= -\frac{y \left(F_{33}\! \left(x , 1\right)-F_{33}\! \left(x , y\right)\right)}{-1+y}\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{64}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{70}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{63}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= -\frac{y \left(F_{18}\! \left(x , 1\right)-F_{18}\! \left(x , y\right)\right)}{-1+y}\\ F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{77}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= y F_{79}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{81}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{79}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{63}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= y F_{91}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{24}\! \left(x \right)+F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{28}\! \left(x \right) F_{98}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{101}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= y x\\ F_{101}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{98}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{95}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)\\ F_{105}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)+F_{113}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= -\frac{-y F_{105}\! \left(x , y\right)+F_{105}\! \left(x , 1\right)}{-1+y}\\ F_{107}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{110}\! \left(x , y\right)\\ F_{108}\! \left(x , y\right) &= F_{107}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= -\frac{F_{33}\! \left(x , 1\right)-F_{33}\! \left(x , y\right)}{-1+y}\\ F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{112}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= -\frac{-F_{16}\! \left(x , y\right) y +F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{31}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\ F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{116}\! \left(x , y\right)+F_{120}\! \left(x , y\right)+F_{24}\! \left(x \right)\\ F_{115}\! \left(x , y\right) &= F_{114}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{117}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{118}\! \left(x , y\right)+F_{119}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= F_{117}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{120}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{81}\! \left(x , y\right)\\ F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)\\ F_{126}\! \left(x , y\right) &= F_{127}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{137}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)+F_{131}\! \left(x , y\right)\\ F_{129}\! \left(x , y\right) &= F_{130}\! \left(x \right) F_{46}\! \left(x , y\right)\\ F_{130}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\ F_{131}\! \left(x , y\right) &= F_{132}\! \left(x , y\right)\\ F_{132}\! \left(x , y\right) &= F_{130}\! \left(x \right) F_{133}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{134}\! \left(x , y\right) &= F_{133}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right)\\ F_{135}\! \left(x , y\right) &= F_{136}\! \left(x \right)+F_{55}\! \left(x , y\right)\\ F_{136}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)\\ F_{138}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{139}\! \left(x , y\right) &= F_{140}\! \left(x , y\right) F_{28}\! \left(x \right)\\ F_{140}\! \left(x , y\right) &= -\frac{-y F_{34}\! \left(x , y\right)+F_{34}\! \left(x , 1\right)}{-1+y}\\ F_{141}\! \left(x , y\right) &= F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\ F_{143}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x , 1\right)\\ F_{145}\! \left(x , y\right) &= y F_{105}\! \left(x , y\right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{2}\! \left(x \right)\\ \end{align*}\)

Av(12354, 12453, 13254, 13452, 14253, 14352, 23154, 24153)

This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 140 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 150 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 142 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion Tracked Component Fusion Symmetries" and has 78 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 148 rules.

Proof Tree

System of Equations