Av(1243, 1432, 2143, 2314)
Generating Function
\(\displaystyle -\frac{x^{2}-3 x +1}{x^{4}+x^{3}-3 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 208, 663, 2113, 6736, 21476, 68472, 218309, 696032, 2219149, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle -\frac{35 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{2687}+\frac{76 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{2687}+\frac{862 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{2687}-\frac{110 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}+Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{2687}\)
This specification was found using the strategy pack "Point Placements" and has 44 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 44 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_{19}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{10}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \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_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \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_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 0\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\
\end{align*}\)