Av(1324, 1342, 1423, 2341, 4123)
Generating Function
\(\displaystyle \frac{-2 \left(x -\frac{1}{2}\right) \left(-1+x \right)^{3} \sqrt{1-4 x}+4 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{2 x \left(2 x -1\right) \left(-1+x \right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 174, 528, 1649, 5328, 17764, 60767, 212029, 751000, 2690718, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(-1+x \right)^{6} F \left(x
\right)^{2}-\left(2 x -1\right) \left(4 x^{4}-7 x^{3}+9 x^{2}-5 x +1\right) \left(-1+x \right)^{3} F \! \left(x \right)+\left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) \left(x^{4}-4 x^{3}+8 x^{2}-5 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +4}-\frac{4 \left(5 n +8\right) a \! \left(n +1\right)}{n +4}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{n +4}+\frac{3 n^{3}-17 n^{2}-4 n +8}{2 n +8}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +4}-\frac{4 \left(5 n +8\right) a \! \left(n +1\right)}{n +4}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{n +4}+\frac{3 n^{3}-17 n^{2}-4 n +8}{2 n +8}, \quad n \geq 7\)
This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 134 rules.
Found on July 23, 2021.Finding the specification took 11 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_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \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_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{54}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{56}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{61}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{72}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{107}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{57}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 3 F_{24}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{9}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{9}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{103}\! \left(x \right) &= 3 F_{24}\! \left(x \right)+F_{104}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x , 1\right)\\
F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right) F_{123}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{133}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{82}\! \left(x \right)\\
F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)+F_{124}\! \left(x , y\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_{9}\! \left(x \right)\\
F_{119}\! \left(x , y\right) &= F_{117}\! \left(x , y\right)+F_{120}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)\\
F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right) F_{123}\! \left(x , y\right)\\
F_{122}\! \left(x , y\right) &= y x\\
F_{123}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{120}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= 2 F_{24}\! \left(x \right)+F_{125}\! \left(x , y\right)+F_{132}\! \left(x , y\right)\\
F_{125}\! \left(x , y\right) &= F_{126}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{126}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\
F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{130}\! \left(x , y\right)+F_{24}\! \left(x \right)\\
F_{128}\! \left(x , y\right) &= F_{122}\! \left(x , y\right) F_{129}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{131}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{116}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{133}\! \left(x , y\right) &= \frac{F_{112}\! \left(x , y\right) y -F_{112}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)