Av(1243, 1324, 1342, 1432, 4213)
Generating Function
\(\displaystyle \frac{x^{8}-3 x^{7}-x^{6}+8 x^{5}-7 x^{4}+12 x^{3}-13 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 149, 394, 1032, 2692, 7013, 18269, 47611, 124148, 323902, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-3 x^{7}-x^{6}+8 x^{5}-7 x^{4}+12 x^{3}-13 x^{2}+6 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 149\)
\(\displaystyle a \! \left(7\right) = 394\)
\(\displaystyle a \! \left(8\right) = 1032\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+3 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right)-2 n -3, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 149\)
\(\displaystyle a \! \left(7\right) = 394\)
\(\displaystyle a \! \left(8\right) = 1032\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+3 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right)-2 n -3, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle -n +\frac{131 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-3 Z^{4}+3 Z^{3}-7 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{590}-\frac{339 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-3 Z^{4}+3 Z^{3}-7 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{590}+\frac{121 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-3 Z^{4}+3 Z^{3}-7 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{295}-\frac{1119 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-3 Z^{4}+3 Z^{3}-7 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{590}+\frac{509 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{5}-3 Z^{4}+3 Z^{3}-7 Z^{2}+5 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{590}+\left(\left\{\begin{array}{cc}2 & n =0 \\ 1 & n =1 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 76 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
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\(\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_{14}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 0\\
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_{40}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{37}\! \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_{11}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{46}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{42}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)