Av(1342, 1432, 3241)
Generating Function
\(\displaystyle -\frac{2 x^{4}-7 x^{3}+8 x^{2}-5 x +1}{x^{5}-6 x^{4}+13 x^{3}-12 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 265, 929, 3249, 11362, 39746, 139060, 486549, 1702349, 5956172, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-6 x^{4}+13 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+2 x^{4}-7 x^{3}+8 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-6 a \! \left(n +1\right)+13 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-6 a \! \left(n +1\right)+13 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{249 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{3089}+\frac{662 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{3089}-\frac{107 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{3089}+\frac{48 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{3089}+\frac{261 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{3089}+\frac{261 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{3089}+\frac{261 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{3089}+\frac{261 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{3089}+\frac{261 \mathit{RootOf} \left(Z^{5}-6 Z^{4}+13 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{3089}\)
This specification was found using the strategy pack "Point Placements" and has 115 rules.
Found on January 18, 2022.Finding the specification took 4 seconds.
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Copy 115 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \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) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \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_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{10}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{10}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{10}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{10}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{10}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{34}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{10}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{10}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= x^{2}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{10}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{10}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{10}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{12}\! \left(x \right)+F_{55}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{10}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{12}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{10}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{10}\! \left(x \right) F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{101}\! \left(x \right) &= 0\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{103}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{104}\! \left(x \right)+F_{108}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{10}\! \left(x \right) F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{10}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{10}\! \left(x \right) F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{112}\! \left(x \right) &= 0\\
F_{113}\! \left(x \right) &= F_{10}\! \left(x \right) F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{64}\! \left(x \right)\\
\end{align*}\)