Av(13452, 13542, 14352, 14523, 14532, 15342, 15423, 15432)
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Counting Sequence
1, 1, 2, 6, 24, 112, 572, 3098, 17486, 101756, 606210, 3679250, 22669600, 141431030, 891659082, ...
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Implicit Equation for the Generating Function
\(\displaystyle -x F \left(x \right)^{4}+\left(4 x -1\right) F \left(x \right)^{3}+\left(x^{3}-6 x^{2}-x +3\right) F \left(x \right)^{2}+\left(7 x^{2}-6 x -3\right) F \! \left(x \right)-x^{2}+4 x +1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 572\)
\(\displaystyle a(7) = 3098\)
\(\displaystyle a(8) = 17486\)
\(\displaystyle a(9) = 101756\)
\(\displaystyle a(10) = 606210\)
\(\displaystyle a(11) = 3679250\)
\(\displaystyle a(12) = 22669600\)
\(\displaystyle a(13) = 141431030\)
\(\displaystyle a(14) = 891659082\)
\(\displaystyle a(15) = 5671965902\)
\(\displaystyle a(16) = 36359236692\)
\(\displaystyle a(17) = 234644815664\)
\(\displaystyle a(18) = 1523242601516\)
\(\displaystyle a(19) = 9940227807740\)
\(\displaystyle a(20) = 65169947154864\)
\(\displaystyle a{\left(n + 21 \right)} = \frac{n \left(n - 1\right) \left(2 n + 1\right) a{\left(n \right)}}{396 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{n \left(41 n^{2} - 14 n - 42\right) a{\left(n + 1 \right)}}{792 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} + \frac{2 \left(n + 21\right) \left(419 n + 7948\right) a{\left(n + 20 \right)}}{99 \left(n + 20\right) \left(n + 22\right)} + \frac{7 \left(4237 n^{2} + 166595 n + 1620807\right) a{\left(n + 18 \right)}}{297 \left(n + 20\right) \left(n + 22\right)} - \frac{2 \left(1186 n^{2} + 54973 n + 605277\right) a{\left(n + 19 \right)}}{297 \left(n + 19\right) \left(n + 22\right)} - \frac{\left(653 n^{3} + 5945 n^{2} + 12556 n + 7210\right) a{\left(n + 2 \right)}}{1584 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} + \frac{\left(11671 n^{3} + 106424 n^{2} + 304427 n + 275212\right) a{\left(n + 3 \right)}}{1584 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(55481 n^{3} + 740336 n^{2} + 2826465 n + 2471174\right) a{\left(n + 6 \right)}}{1584 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(65303 n^{3} + 723379 n^{2} + 2623060 n + 3114324\right) a{\left(n + 4 \right)}}{1584 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(298108 n^{3} + 14994075 n^{2} + 251763896 n + 1412237700\right) a{\left(n + 17 \right)}}{594 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} + \frac{\left(470423 n^{3} + 6176751 n^{2} + 26670280 n + 37835196\right) a{\left(n + 5 \right)}}{4752 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(667099 n^{3} + 15599747 n^{2} + 117901668 n + 286275460\right) a{\left(n + 9 \right)}}{792 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(1045231 n^{3} + 20370930 n^{2} + 133904111 n + 296384400\right) a{\left(n + 7 \right)}}{2376 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(1719227 n^{3} + 87142332 n^{2} + 1468780627 n + 8232442482\right) a{\left(n + 16 \right)}}{1188 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(2459762 n^{3} + 106208909 n^{2} + 1521116616 n + 7222709568\right) a{\left(n + 15 \right)}}{396 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} + \frac{\left(4852235 n^{3} + 108671070 n^{2} + 817321123 n + 2066379984\right) a{\left(n + 8 \right)}}{4752 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} + \frac{\left(5205497 n^{3} + 176789862 n^{2} + 1997051891 n + 7504554740\right) a{\left(n + 11 \right)}}{792 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(12620719 n^{3} + 407460498 n^{2} + 4374011561 n + 15594442542\right) a{\left(n + 10 \right)}}{4752 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} + \frac{\left(16687087 n^{3} + 692289501 n^{2} + 9548443157 n + 43786973823\right) a{\left(n + 14 \right)}}{1188 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(16953187 n^{3} + 571965300 n^{2} + 6397354217 n + 23718000960\right) a{\left(n + 12 \right)}}{4752 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)} - \frac{\left(43659779 n^{3} + 1740959781 n^{2} + 23084352094 n + 101803244856\right) a{\left(n + 13 \right)}}{4752 \left(n + 19\right) \left(n + 20\right) \left(n + 22\right)}, \quad n \geq 21\)
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This specification was found using the strategy pack "Point Placements Req Corrob" and has 97 rules.

Finding the specification took 45189 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_{18}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{17}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{4} \left(x \right)^{2} F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{0}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{0}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{38}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{21}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{44}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{0}\! \left(x \right) F_{18}\! \left(x \right) F_{27}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{18}\! \left(x \right) F_{50}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{51}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{0}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{18}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{13}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{19}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{18}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{68}\! \left(x \right) &= -F_{72}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= \frac{F_{70}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{18}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= \frac{F_{85}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{18}\! \left(x \right) F_{4}\! \left(x \right)}\\ F_{94}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right) F_{93}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 97 rules.

Finding the specification took 14766 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_{18}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{17}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{4} \left(x \right)^{2} F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{0}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{0} \left(x \right)^{2} F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{0}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{9}\! \left(x \right)}\\ F_{38}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{44}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{0}\! \left(x \right) F_{18}\! \left(x \right) F_{27}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{18}\! \left(x \right) F_{49}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{50}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{0}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{18}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{19}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{18}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{67}\! \left(x \right) &= -F_{72}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{18}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= \frac{F_{85}\! \left(x \right)}{F_{18}\! \left(x \right)}\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{93}\! \left(x \right) &= \frac{F_{94}\! \left(x \right)}{F_{18}\! \left(x \right) F_{4}\! \left(x \right)}\\ F_{94}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right) F_{93}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 183 rules.

Finding the specification took 30401 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_{53}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{178}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{177}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= y x\\ F_{18}\! \left(x , y\right) &= y F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{34}\! \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_{1}\! \left(x \right)+F_{32}\! \left(x , y\right)\\ F_{36}\! \left(x , y\right) &= F_{17}\! \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_{17}\! \left(x , y\right) F_{32}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\ F_{40}\! \left(x \right) &= 0\\ F_{41}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{50}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{53}\! \left(x \right) &= x\\ F_{55}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{53}\! \left(x \right) F_{55}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{61}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{174}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{171}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{169}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{53}\! \left(x \right) F_{65}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{70}\! \left(x \right)+F_{75}\! \left(x , y\right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{53}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{75}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{76}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{53}\! \left(x \right) F_{77}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{81}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{82}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{53}\! \left(x \right)+F_{79}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{86}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= y \,x^{2}\\ F_{87}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{53}\! \left(x \right)\\ F_{89}\! \left(x , y\right) &= F_{40}\! \left(x \right)+F_{90}\! \left(x , y\right)+F_{92}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{53}\! \left(x \right) F_{91}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= y F_{79}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{93}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= 0\\ F_{95}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{157}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{2}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{104}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= \frac{F_{106}\! \left(x \right)}{F_{53}\! \left(x \right)}\\ F_{106}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{105}\! \left(x \right) F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{118}\! \left(x \right) &= \frac{F_{119}\! \left(x \right)}{F_{53}\! \left(x \right)}\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{121}\! \left(x \right) &= \frac{F_{122}\! \left(x \right)}{F_{53}\! \left(x \right)}\\ F_{122}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{118}\! \left(x \right) F_{125}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{118}\! \left(x \right) F_{128}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{129}\! \left(x \right) &= -F_{133}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= \frac{F_{131}\! \left(x \right)}{F_{53}\! \left(x \right)}\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{104}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{135}\! \left(x \right) &= \frac{F_{136}\! \left(x \right)}{F_{53}\! \left(x \right)}\\ F_{136}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{2}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{144}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{145}\! \left(x , y\right) &= F_{144}\! \left(x , y\right)+F_{156}\! \left(x , y\right)\\ F_{146}\! \left(x , y\right) &= F_{145}\! \left(x , y\right)+F_{153}\! \left(x \right)\\ F_{146}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{147}\! \left(x , y\right)\\ F_{147}\! \left(x , y\right) &= F_{148}\! \left(x , y\right)\\ F_{148}\! \left(x , y\right) &= F_{149}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{149}\! \left(x , y\right) &= F_{150}\! \left(x , y\right)+F_{151}\! \left(x , y\right)\\ F_{150}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{151}\! \left(x , y\right) &= F_{152}\! \left(x , y\right)\\ F_{152}\! \left(x , y\right) &= F_{53}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{156}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\ F_{157}\! \left(x , y\right) &= F_{158}\! \left(x , y\right)\\ F_{158}\! \left(x , y\right) &= F_{159}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{159}\! \left(x , y\right) &= F_{160}\! \left(x , y\right)+F_{163}\! \left(x , y\right)\\ F_{160}\! \left(x , y\right) &= F_{161}\! \left(x , y\right) F_{2}\! \left(x \right)\\ F_{162}\! \left(x , y\right) &= F_{161}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{162}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)\\ F_{163}\! \left(x , y\right) &= F_{164}\! \left(x , y\right)\\ F_{164}\! \left(x , y\right) &= F_{165}\! \left(x , y\right)+F_{167}\! \left(x , y\right)\\ F_{165}\! \left(x , y\right) &= F_{166}\! \left(x , y\right)\\ F_{166}\! \left(x , y\right) &= F_{118}\! \left(x \right) F_{151}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{167}\! \left(x , y\right) &= F_{168}\! \left(x , y\right)\\ F_{168}\! \left(x , y\right) &= F_{118}\! \left(x \right) F_{19}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{169}\! \left(x , y\right) &= F_{170}\! \left(x , y\right)\\ F_{170}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{53}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{171}\! \left(x , y\right) &= F_{172}\! \left(x , y\right)\\ F_{172}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{21}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{173}\! \left(x , y\right) &= y F_{171}\! \left(x , y\right)\\ F_{174}\! \left(x , y\right) &= y F_{175}\! \left(x , y\right)\\ F_{175}\! \left(x , y\right) &= F_{176}\! \left(x , y\right)\\ F_{176}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{53}\! \left(x \right) F_{63}\! \left(x , y\right)\\ F_{177}\! \left(x , y\right) &= y F_{151}\! \left(x , y\right)\\ F_{178}\! \left(x , y\right) &= F_{179}\! \left(x , y\right)\\ F_{179}\! \left(x , y\right) &= F_{180}\! \left(x , y\right) F_{53}\! \left(x \right)\\ F_{181}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{180}\! \left(x , y\right)\\ F_{181}\! \left(x , y\right) &= F_{182}\! \left(x , y\right)\\ F_{182}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\ \end{align*}\)