Av(1234, 1243, 2134, 2431, 4213)
Generating Function
\(\displaystyle \frac{2 x^{15}+2 x^{14}-10 x^{13}-14 x^{12}+10 x^{11}+35 x^{10}+12 x^{9}-36 x^{8}-16 x^{7}+12 x^{6}+10 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 49, 105, 213, 435, 877, 1748, 3448, 6740, 13072, 25182, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{15}+2 x^{14}-10 x^{13}-14 x^{12}+10 x^{11}+35 x^{10}+12 x^{9}-36 x^{8}-16 x^{7}+12 x^{6}+10 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 105\)
\(\displaystyle a \! \left(7\right) = 213\)
\(\displaystyle a \! \left(8\right) = 435\)
\(\displaystyle a \! \left(9\right) = 877\)
\(\displaystyle a \! \left(10\right) = 1748\)
\(\displaystyle a \! \left(11\right) = 3448\)
\(\displaystyle a \! \left(12\right) = 6740\)
\(\displaystyle a \! \left(13\right) = 13072\)
\(\displaystyle a \! \left(14\right) = 25182\)
\(\displaystyle a \! \left(15\right) = 48230\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}-n +\frac{5}{2}, \quad n \geq 16\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 105\)
\(\displaystyle a \! \left(7\right) = 213\)
\(\displaystyle a \! \left(8\right) = 435\)
\(\displaystyle a \! \left(9\right) = 877\)
\(\displaystyle a \! \left(10\right) = 1748\)
\(\displaystyle a \! \left(11\right) = 3448\)
\(\displaystyle a \! \left(12\right) = 6740\)
\(\displaystyle a \! \left(13\right) = 13072\)
\(\displaystyle a \! \left(14\right) = 25182\)
\(\displaystyle a \! \left(15\right) = 48230\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}-n +\frac{5}{2}, \quad n \geq 16\)
Explicit Closed Form
\(\displaystyle \frac{14 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{11}+\frac{43 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{22}+\frac{53 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}+Z^{2}+Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{22}+\frac{11 \left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{5}+2 \left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)-\frac{11 \left(n +1\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -2}\right)}{5}+\frac{67 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{25}+\frac{14 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{25}-\frac{156 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{3}-2 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{25}+\left(\left\{\begin{array}{cc}3 & n =0 \\ 1 & n =1 \\ -2 & n =2 \\ -4 & n =3\text{ or } n =4 \\ 2 & n =6 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 80 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 80 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_{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_{18}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \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_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
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_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \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_{12}\! \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_{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_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \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_{12}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{58}\! \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_{25}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{71}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{72}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
\end{align*}\)