Av(1243, 1432, 4231)
Generating Function
\(\displaystyle -\frac{2 x^{6}-10 x^{5}+25 x^{4}-29 x^{3}+20 x^{2}-7 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 238, 726, 2101, 5857, 15926, 42626, 112997, 297861, 782666, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{6}-10 x^{5}+25 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 238\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{n \left(n +1\right) \left(n^{2}+n +4\right)}{12}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 238\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+\frac{n \left(n +1\right) \left(n^{2}+n +4\right)}{12}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{\left(-24 \sqrt{5}+120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(24 \sqrt{5}+120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{60}-\frac{n^{4}}{12}+\frac{n^{3}}{6}-\frac{17 n^{2}}{12}+\frac{4 n}{3}-3\)
This specification was found using the strategy pack "Insertion Row Placements" and has 152 rules.
Found on July 23, 2021.Finding the specification took 7 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{41}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{42}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{46}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{15}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{59}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{66}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{67}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{71}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{8}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{8}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{89}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{10}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{93}\! \left(x \right)+F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{8}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{8}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{93}\! \left(x \right)+F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{77}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{15}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{108}\! \left(x \right)+F_{15}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{101}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{102}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{112}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{113}\! \left(x \right)+F_{114}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{113}\! \left(x \right) &= 0\\
F_{114}\! \left(x \right) &= F_{105}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{109}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{119}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{120}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{11}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{147}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{149}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{140}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{8}\! \left(x \right) F_{81}\! \left(x \right)\\
\end{align*}\)