Av(1234, 1243, 2431)
Generating Function
\(\displaystyle \frac{x^{9}-4 x^{8}+20 x^{6}-58 x^{5}+83 x^{4}-69 x^{3}+34 x^{2}-9 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 256, 826, 2535, 7474, 21370, 59718, 164082, 445266, 1197326, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(-1+x \right)^{5} F \! \left(x \right)+x^{9}-4 x^{8}+20 x^{6}-58 x^{5}+83 x^{4}-69 x^{3}+34 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 256\)
\(\displaystyle a \! \left(7\right) = 826\)
\(\displaystyle a \! \left(8\right) = 2535\)
\(\displaystyle a \! \left(9\right) = 7474\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{4}}{24}+\frac{7 n^{3}}{12}-\frac{11 n^{2}}{24}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{23 n}{12}+2, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 256\)
\(\displaystyle a \! \left(7\right) = 826\)
\(\displaystyle a \! \left(8\right) = 2535\)
\(\displaystyle a \! \left(9\right) = 7474\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{4}}{24}+\frac{7 n^{3}}{12}-\frac{11 n^{2}}{24}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{23 n}{12}+2, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-36 \sqrt{5}+300\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(36 \sqrt{5}+300\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}-\frac{n^{4}}{24}+\frac{7 n^{3}}{12}\\-\frac{35 n^{2}}{24}+\frac{95 n}{12}-\frac{25 \,2^{n}}{4} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 175 rules.
Found on January 18, 2022.Finding the specification took 20 seconds.
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Copy 175 equations to clipboard:
\(\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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{19}\! \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_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{172}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{142}\! \left(x \right)+F_{169}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 0\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{139}\! \left(x \right)+F_{49}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \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_{52}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{61}\! \left(x \right) &= 0\\
F_{62}\! \left(x \right) &= 0\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 3 F_{13}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 0\\
F_{78}\! \left(x \right) &= 0\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{83}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{71}\! \left(x \right)+F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{92}\! \left(x \right) &= 0\\
F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{100}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{108}\! \left(x \right)+F_{112}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{112}\! \left(x \right) &= 0\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{120}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{112}\! \left(x \right)+F_{113}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{146}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{142}\! \left(x \right)+F_{151}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{34}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{157}\! \left(x \right)+F_{71}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{163}\! \left(x \right)+F_{167}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{160}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{167}\! \left(x \right) &= 0\\
F_{168}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{2}\! \left(x \right)\\
\end{align*}\)