Av(1243, 2341, 4231)
Generating Function
\(\displaystyle \frac{x^{9}-26 x^{8}+138 x^{7}-317 x^{6}+419 x^{5}-350 x^{4}+188 x^{3}-63 x^{2}+12 x -1}{\left(2 x -1\right)^{4} \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 21, 74, 248, 784, 2355, 6785, 18897, 51177, 135358, 350788, 893038, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{4} \left(x -1\right)^{5} F \! \left(x \right)-x^{9}+26 x^{8}-138 x^{7}+317 x^{6}-419 x^{5}+350 x^{4}-188 x^{3}+63 x^{2}-12 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(6\right) = 248\)
\(\displaystyle a \! \left(7\right) = 784\)
\(\displaystyle a \! \left(8\right) = 2355\)
\(\displaystyle a \! \left(9\right) = 6785\)
\(\displaystyle a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)-\frac{\left(n -2\right) \left(n -3\right) \left(n^{2}-9 n -16\right)}{24}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(6\right) = 248\)
\(\displaystyle a \! \left(7\right) = 784\)
\(\displaystyle a \! \left(8\right) = 2355\)
\(\displaystyle a \! \left(9\right) = 6785\)
\(\displaystyle a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)-\frac{\left(n -2\right) \left(n -3\right) \left(n^{2}-9 n -16\right)}{24}, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(3 n^{3}-24 n^{2}+123 n -6\right) 2^{n}}{96}-\frac{n^{4}}{24}-\frac{n^{3}}{12}-\frac{11 n^{2}}{24}-\frac{17 n}{12}+1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 194 rules.
Found on January 18, 2022.Finding the specification took 6 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \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_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{25}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{56}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{65}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{12}\! \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_{67}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{78}\! \left(x \right)+F_{81}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{12}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{12}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{12}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{100}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{108}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{43}\! \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_{12}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{12}\! \left(x \right) F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{107}\! \left(x \right)\\
F_{124}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{125}\! \left(x \right)+F_{141}\! \left(x \right)+F_{145}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{12}\! \left(x \right) F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= 2 F_{25}\! \left(x \right)+F_{129}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{12}\! \left(x \right) F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{12}\! \left(x \right) F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= 3 F_{25}\! \left(x \right)+F_{135}\! \left(x \right)+F_{139}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{12}\! \left(x \right) F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{12}\! \left(x \right) F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{12}\! \left(x \right) F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{106}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{12}\! \left(x \right) F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{153}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{12}\! \left(x \right) F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{12}\! \left(x \right) F_{148}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{160}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{12}\! \left(x \right) F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{12}\! \left(x \right) F_{161}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{12}\! \left(x \right) F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{163}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{12}\! \left(x \right) F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{173}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{163}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{169}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{179}\! \left(x \right)+F_{193}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{12}\! \left(x \right) F_{176}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{12}\! \left(x \right) F_{180}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)+F_{187}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{182}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{12}\! \left(x \right) F_{184}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{186}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{182}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{12}\! \left(x \right) F_{190}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right)+F_{192}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{182}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{188}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{12}\! \left(x \right) F_{154}\! \left(x \right)\\
\end{align*}\)