Av(1243, 1342, 1423, 2134, 2413)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x^{3}+2 x -1\right)}{\left(3 x -1\right) \left(x^{3}-x^{2}+2 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 175, 527, 1585, 4762, 14298, 42915, 128782, 386411, 1159347, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -1\right) \left(x^{3}-x^{2}+2 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{3}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -3 a \! \left(n \right)+4 a \! \left(n +1\right)-7 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -3 a \! \left(n \right)+4 a \! \left(n +1\right)-7 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{1071 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-4 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{2783}+\frac{1412 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-4 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{2783}-\frac{1966 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-4 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{2783}+\frac{763 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{4}-4 Z^{3}+7 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{2783}+\left(\left\{\begin{array}{cc}\frac{2}{3} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 63 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 63 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_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{54}\! \left(x \right)\\
\end{align*}\)