Av(1342, 2413, 3412, 4123)
Generating Function
\(\displaystyle -\frac{5 x^{4}-12 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{4}-6 x^{3}+8 x^{2}-5 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 203, 617, 1847, 5484, 16207, 47756, 140437, 412407, 1209892, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+5 x^{4}-12 x^{3}+13 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{8576 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-13 Z^{4}+22 Z^{3}-18 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{5067}-\frac{17222 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-13 Z^{4}+22 Z^{3}-18 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{1689}+\frac{70177 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-13 Z^{4}+22 Z^{3}-18 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{5067}-\frac{43177 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-13 Z^{4}+22 Z^{3}-18 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{5067}+\frac{9122 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-13 Z^{4}+22 Z^{3}-18 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{5067}\)
This specification was found using the strategy pack "Point Placements" and has 51 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 51 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \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_{13}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \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_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{13}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{17}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{13}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{43}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{13}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{43}\! \left(x \right)\\
\end{align*}\)