Av(1234, 1243, 3214, 3241)
Generating Function
\(\displaystyle -\frac{2 x^{8}+x^{7}-7 x^{6}-2 x^{5}+4 x^{4}+6 x^{3}-3 x +1}{\left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 60, 166, 446, 1160, 2958, 7428, 18452, 45474, 111426, 271892, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+2 x -1\right) \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{8}+x^{7}-7 x^{6}-2 x^{5}+4 x^{4}+6 x^{3}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(7\right) = 446\)
\(\displaystyle a \! \left(8\right) = 1160\)
\(\displaystyle a \! \left(n +6\right) = a \! \left(n \right)+4 a \! \left(n +1\right)+2 a \! \left(n +2\right)-6 a \! \left(n +3\right)-2 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(7\right) = 446\)
\(\displaystyle a \! \left(8\right) = 1160\)
\(\displaystyle a \! \left(n +6\right) = a \! \left(n \right)+4 a \! \left(n +1\right)+2 a \! \left(n +2\right)-6 a \! \left(n +3\right)-2 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{\left(\left\{\begin{array}{cc}2 & n =0 \\ 2 & n =1 \\ 4 & n =2 \\ -11 \left(-1-\sqrt{2}\right)^{-n -1}-11 \left(\sqrt{2}-1\right)^{-n -1}+\\\frac{\left(\left(-2 n +4\right) \sqrt{5}-10 n \right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+\\\frac{\left(\left(2 n -4\right) \sqrt{5}-10 n \right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5}+29 \left(-1-\sqrt{2}\right)^{-n}+29 \left(\sqrt{2}-1\right)^{-n} & \text{otherwise} \end{array}\right.\right)}{2}\)
This specification was found using the strategy pack "Point Placements" and has 43 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 43 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \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) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 0\\
F_{42}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)