Av(1234, 2413, 2431, 3142, 4213)
Generating Function
\(\displaystyle -\frac{x^{11}+3 x^{10}+4 x^{9}-4 x^{8}-13 x^{7}-x^{6}+6 x^{5}+6 x^{4}-8 x^{2}+5 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 51, 120, 264, 557, 1145, 2315, 4627, 9169, 18045, 35305, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{11}+3 x^{10}+4 x^{9}-4 x^{8}-13 x^{7}-x^{6}+6 x^{5}+6 x^{4}-8 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 120\)
\(\displaystyle a \! \left(7\right) = 264\)
\(\displaystyle a \! \left(8\right) = 557\)
\(\displaystyle a \! \left(9\right) = 1145\)
\(\displaystyle a \! \left(10\right) = 2315\)
\(\displaystyle a \! \left(11\right) = 4627\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{2}}{4}+\frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+2 n -\frac{11}{2}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 120\)
\(\displaystyle a \! \left(7\right) = 264\)
\(\displaystyle a \! \left(8\right) = 557\)
\(\displaystyle a \! \left(9\right) = 1145\)
\(\displaystyle a \! \left(10\right) = 2315\)
\(\displaystyle a \! \left(11\right) = 4627\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{2}}{4}+\frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+2 n -\frac{11}{2}, \quad n \geq 12\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(19525 \,\mathrm{I}-4875 \sqrt{11}\right) \sqrt{3}-14625 \,\mathrm{I} \sqrt{11}+19525\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+57200+\left(\left(37675 \,\mathrm{I}+6075 \sqrt{11}\right) \sqrt{3}-18225 \,\mathrm{I} \sqrt{11}-37675\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(\left(-37675 \,\mathrm{I}+6075 \sqrt{11}\right) \sqrt{3}+18225 \,\mathrm{I} \sqrt{11}-37675\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+57200+\left(\left(-19525 \,\mathrm{I}-4875 \sqrt{11}\right) \sqrt{3}+14625 \,\mathrm{I} \sqrt{11}+19525\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(-12150 \sqrt{11}\, \sqrt{3}+75350\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+9750 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-39050 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+57200\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{13200}\\+\frac{\left(\left(6600 n +12936\right) \sqrt{5}-14520 n -33000\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}+\\\frac{\left(\left(-6600 n -12936\right) \sqrt{5}-14520 n -33000\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}-\frac{n^{2}}{2}-6 n -4 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 62 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 62 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_{27}\! \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_{21}\! \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_{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_{14}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \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_{36}\! \left(x \right) &= F_{28}\! \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_{28}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{41}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \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_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{38}\! \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_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{51}\! \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_{61}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{53}\! \left(x \right)\\
\end{align*}\)