Av(1243, 1342, 1423, 2134, 2143)
Generating Function
\(\displaystyle \frac{x^{3}+x^{2}+2 x -1}{x^{3}+3 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 183, 568, 1763, 5472, 16984, 52715, 163617, 507835, 1576220, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+3 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\left(2 \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \sqrt{5}\, \left(\frac{\left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{4}+\mathrm{I} \sqrt{3}-1\right)^{-n} \left(\frac{\left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}+1\right)^{\frac{1}{3}}+\left(2 \,\mathrm{I} \sqrt{3}-2\right) 2^{\frac{1}{3}}}{\left(-8 \sqrt{5}+8\right) \left(\sqrt{5}+1\right)^{\frac{1}{3}}+16 \,2^{\frac{1}{3}}}\right)^{n}+10 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}} \left(2 \left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \sqrt{3}-8\right)^{-n} \left(\frac{\left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}+1\right)^{\frac{1}{3}}+\left(2 \,\mathrm{I} \sqrt{3}-2\right) 2^{\frac{1}{3}}}{\left(-\sqrt{5}+1\right) \left(\sqrt{5}+1\right)^{\frac{1}{3}}+2 \,2^{\frac{1}{3}}}\right)^{n}+\left(\left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{5}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(\sqrt{5}-\frac{5}{3}\right)}{2}\right)\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}+1\right)^{\frac{2}{3}} 2^{\frac{1}{3}} \left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right)}{8}\right)^{n}-\left(\left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}+1\right)^{\frac{2}{3}} 2^{\frac{1}{3}} \left(\sqrt{5}-1\right)}{8}\right)^{n} \left(\left(1+\mathrm{I} \sqrt{3}\right) \left(5+\sqrt{5}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(\sqrt{5}-\frac{5}{3}\right)}{2}\right)-5 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{16}-\frac{\left(\sqrt{5}+1\right)^{\frac{2}{3}} 2^{\frac{1}{3}} \left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right)}{16}\right)^{n} \left(\frac{\left(\sqrt{5}-1\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}+1\right)^{\frac{1}{3}}+\left(2 \,\mathrm{I} \sqrt{3}-2\right) 2^{\frac{1}{3}}}{\left(-\sqrt{5}+1\right) \left(\sqrt{5}+1\right)^{\frac{1}{3}}+2 \,2^{\frac{1}{3}}}\right)^{n}\right)\right)}{120} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 60 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 60 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_{21}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{39}\! \left(x \right) &= 0\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{36}\! \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_{42}\! \left(x \right)\\
F_{45}\! \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_{36}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \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_{2}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{47}\! \left(x \right)\\
\end{align*}\)