Av(1243, 2314, 2413, 3142)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}{3 x^{6}-17 x^{5}+44 x^{4}-52 x^{3}+31 x^{2}-9 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 221, 719, 2320, 7457, 23936, 76817, 246581, 791749, 2542839, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{6}-17 x^{5}+44 x^{4}-52 x^{3}+31 x^{2}-9 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+17 a \! \left(n +1\right)-44 a \! \left(n +2\right)+52 a \! \left(n +3\right)-31 a \! \left(n +4\right)+9 a \! \left(n +5\right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+17 a \! \left(n +1\right)-44 a \! \left(n +2\right)+52 a \! \left(n +3\right)-31 a \! \left(n +4\right)+9 a \! \left(n +5\right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{132009 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +4}}{93253}-\frac{132009 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +4}}{93253}-\frac{132009 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +4}}{93253}-\frac{132009 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +4}}{93253}-\frac{132009 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +4}}{93253}-\frac{132009 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +4}}{93253}+\frac{671623 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +3}}{93253}+\frac{671623 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +3}}{93253}+\frac{671623 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +3}}{93253}+\frac{671623 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +3}}{93253}+\frac{671623 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +3}}{93253}+\frac{671623 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +3}}{93253}-\frac{1552642 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +2}}{93253}-\frac{1552642 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +2}}{93253}-\frac{1552642 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +2}}{93253}-\frac{1552642 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +2}}{93253}-\frac{1552642 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +2}}{93253}-\frac{1552642 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +2}}{93253}+\frac{1414429 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +1}}{93253}+\frac{1414429 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +1}}{93253}+\frac{1414429 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +1}}{93253}+\frac{1414429 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +1}}{93253}+\frac{1414429 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +1}}{93253}+\frac{1414429 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +1}}{93253}+\frac{88782 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n -1}}{93253}+\frac{88782 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n -1}}{93253}+\frac{88782 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n -1}}{93253}+\frac{88782 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n -1}}{93253}+\frac{88782 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n -1}}{93253}+\frac{88782 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n -1}}{93253}-\frac{575169 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n}}{93253}-\frac{575169 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n}}{93253}-\frac{575169 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n}}{93253}-\frac{575169 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n}}{93253}-\frac{575169 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n}}{93253}-\frac{575169 \mathit{RootOf} \left(3 Z^{6}-17 Z^{5}+44 Z^{4}-52 Z^{3}+31 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n}}{93253}\)
This specification was found using the strategy pack "Point Placements" and has 56 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 56 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \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_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
\end{align*}\)