Av(1243, 1324, 2143, 2341, 2413)
Generating Function
\(\displaystyle \frac{x^{7}-x^{6}-x^{5}-2 x^{4}-7 x^{3}+12 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 151, 404, 1067, 2798, 7308, 19047, 49593, 129083, 336000, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-x^{7}+x^{6}+x^{5}+2 x^{4}+7 x^{3}-12 x^{2}+6 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 151\)
\(\displaystyle a \! \left(7\right) = 404\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-4 a \! \left(n +2\right)-3 a \! \left(n +3\right)+16 a \! \left(n +4\right)-17 a \! \left(n +5\right)+7 a \! \left(n +6\right), \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 151\)
\(\displaystyle a \! \left(7\right) = 404\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-4 a \! \left(n +2\right)-3 a \! \left(n +3\right)+16 a \! \left(n +4\right)-17 a \! \left(n +5\right)+7 a \! \left(n +6\right), \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{342 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{575}-\frac{994 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{575}+\frac{18 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{115}+\frac{1651 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{575}-\frac{729 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{115}+\frac{914 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{575}+\frac{194 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{7}-4 Z^{6}+4 Z^{5}+3 Z^{4}-16 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{575}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 73 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 73 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_{23}\! \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_{21}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{40}\! \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_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 0\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{52}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\
\end{align*}\)