Av(1342, 1432, 2314, 2431, 4123)
Generating Function
\(\displaystyle \frac{2 x^{12}-5 x^{11}+2 x^{10}+2 x^{9}+x^{8}+12 x^{7}-23 x^{6}+18 x^{5}-21 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 117, 258, 550, 1149, 2366, 4819, 9733, 19532, 39011, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{12}-5 x^{11}+2 x^{10}+2 x^{9}+x^{8}+12 x^{7}-23 x^{6}+18 x^{5}-21 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 117\)
\(\displaystyle a \! \left(7\right) = 258\)
\(\displaystyle a \! \left(8\right) = 550\)
\(\displaystyle a \! \left(9\right) = 1149\)
\(\displaystyle a \! \left(10\right) = 2366\)
\(\displaystyle a \! \left(11\right) = 4819\)
\(\displaystyle a \! \left(12\right) = 9733\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{4}}{12}+\frac{5 n^{3}}{6}-\frac{35 n^{2}}{12}-2 a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right)-\frac{5 n}{6}-1, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 117\)
\(\displaystyle a \! \left(7\right) = 258\)
\(\displaystyle a \! \left(8\right) = 550\)
\(\displaystyle a \! \left(9\right) = 1149\)
\(\displaystyle a \! \left(10\right) = 2366\)
\(\displaystyle a \! \left(11\right) = 4819\)
\(\displaystyle a \! \left(12\right) = 9733\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{4}}{12}+\frac{5 n^{3}}{6}-\frac{35 n^{2}}{12}-2 a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right)-\frac{5 n}{6}-1, \quad n \geq 13\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(\left(44 \,\mathrm{I}-12 \sqrt{11}\right) \sqrt{3}-36 \,\mathrm{I} \sqrt{11}+44\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+88+\left(\left(110 \,\mathrm{I}+18 \sqrt{11}\right) \sqrt{3}-54 \,\mathrm{I} \sqrt{11}-110\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}}{264}\\+\\\frac{\left(\left(\left(-110 \,\mathrm{I}+18 \sqrt{11}\right) \sqrt{3}+54 \,\mathrm{I} \sqrt{11}-110\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+88+\left(\left(-44 \,\mathrm{I}-12 \sqrt{11}\right) \sqrt{3}+36 \,\mathrm{I} \sqrt{11}+44\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}}{264}\\+\\\frac{\left(\left(-36 \sqrt{11}\, \sqrt{3}+220\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+24 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-88 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+88\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}}{264}\\-\frac{n^{4}}{24}+\frac{n^{3}}{4}-\frac{35 n^{2}}{24}-\frac{7 n}{4}+2^{n +1}-5 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 78 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 78 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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \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_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \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_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
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_{35}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \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_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{36}\! \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_{43}\! \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_{4}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{51}\! \left(x \right)+F_{75}\! \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_{61}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= x^{2}\\
F_{61}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{44}\! \left(x \right)\\
\end{align*}\)