Av(1234, 1243, 1432, 2341, 3142)
Generating Function
\(\displaystyle \frac{x^{9}-2 x^{8}+2 x^{7}+2 x^{6}-5 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 140, 366, 948, 2431, 6179, 15590, 39091, 97498, 242051, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+x^{9}-2 x^{8}+2 x^{7}+2 x^{6}-5 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 140\)
\(\displaystyle a \! \left(7\right) = 366\)
\(\displaystyle a \! \left(8\right) = 948\)
\(\displaystyle a \! \left(9\right) = 2431\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 140\)
\(\displaystyle a \! \left(7\right) = 366\)
\(\displaystyle a \! \left(8\right) = 948\)
\(\displaystyle a \! \left(9\right) = 2431\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \frac{\left(-92 n -92\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -2}\right)}{529}+\frac{\left(230 n -450\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{529}-\frac{687 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{529}+\frac{\left(23 n +1943\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{529}+\left(\left\{\begin{array}{cc}-4 & n =0 \\ 2 & n =2 \\ 1 & n =3 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 72 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
Copy 72 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_{19}\! \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_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \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_{55}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \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_{42}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{71}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
\end{align*}\)