Av(1432, 2413, 3412)
Generating Function
\(\displaystyle \frac{x^{3}-3 x^{2}+4 x -1}{3 x^{3}-6 x^{2}+5 x -1}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 267, 948, 3363, 11928, 42306, 150051, 532203, 1887627, 6695070, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{3}-6 x^{2}+5 x -1\right) F \! \left(x \right)-x^{3}+3 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 3 a \! \left(n \right)-6 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 3 a \! \left(n \right)-6 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{29}+29\right) 2^{\frac{2}{3}} \left(5+\sqrt{29}\right)^{\frac{1}{3}}+464-17 \,2^{\frac{1}{3}} \left(\sqrt{29}-\frac{87}{17}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}+2\right) \left(20+4 \sqrt{29}\right)^{\frac{1}{3}}}{24}+\frac{2}{3}-\frac{\left(\sqrt{29}-5\right) 2^{\frac{1}{3}} \left(1+\mathrm{I} \sqrt{3}\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2088}\\+\\\frac{\left(\left(\sqrt{29}+29\right) \left(1+\mathrm{I} \sqrt{3}\right) 2^{\frac{2}{3}} \left(5+\sqrt{29}\right)^{\frac{1}{3}}+464+17 \,2^{\frac{1}{3}} \left(\sqrt{29}-\frac{87}{17}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}+2\right) \left(20+4 \sqrt{29}\right)^{\frac{1}{3}}}{24}+\frac{2}{3}+\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{29}-5\right) 2^{\frac{1}{3}} \left(5+\sqrt{29}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2088}\\+\\\frac{17 \left(-\frac{2^{\frac{2}{3}} \left(\sqrt{29}+29\right) \left(5+\sqrt{29}\right)^{\frac{1}{3}}}{17}+\frac{232}{17}+2^{\frac{1}{3}} \left(\sqrt{29}-\frac{87}{17}\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}\right) \left(-\frac{\left(20+4 \sqrt{29}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}+\frac{2^{\frac{1}{3}} \left(\sqrt{29}-5\right) \left(5+\sqrt{29}\right)^{\frac{2}{3}}}{12}\right)^{-n}}{1044} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 65 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 65 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{58}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\
\end{align*}\)