Av(1234, 1342, 2143, 2413, 4213)
Generating Function
\(\displaystyle \frac{x^{8}-5 x^{6}+5 x^{5}-3 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 52, 136, 349, 880, 2185, 5359, 13018, 31384, 75202, 179311, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+x^{8}-5 x^{6}+5 x^{5}-3 x^{4}+6 x^{3}-9 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 136\)
\(\displaystyle a \! \left(7\right) = 349\)
\(\displaystyle a \! \left(8\right) = 880\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+1, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 136\)
\(\displaystyle a \! \left(7\right) = 349\)
\(\displaystyle a \! \left(8\right) = 880\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+1, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{1181 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+13 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{3-n}\right)}{46}-\frac{3563 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+13 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{46}+\frac{5867 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+13 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{46}-\frac{2305 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+13 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{23}+\frac{1171 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-7 Z^{4}+13 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{46}+\frac{\left(\left\{\begin{array}{cc}-\frac{69}{8} & n =0 \\ \frac{3}{4} & n =1 \\ \frac{7}{2} & n =2 \\ 1 & n =3 \\ 0 & \text{otherwise} \end{array}\right.\right)}{2}\)
This specification was found using the strategy pack "Point Placements" and has 88 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
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Copy 88 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_{24}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 0\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \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_{4}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{59}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{46}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{38}\! \left(x \right)+F_{66}\! \left(x \right)+F_{77}\! \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_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{79}\! \left(x \right) &= 2 F_{38}\! \left(x \right)+F_{66}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{73}\! \left(x \right)\\
\end{align*}\)