Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 23145, 23154, 24135, 24153, 25134, 25143)
Generating Function
\(\displaystyle \frac{5 x^{3}-x^{2}+3 x -1}{6 x^{3}-3 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 90, 324, 1170, 4248, 15426, 55980, 203130, 737136, 2675034, 9707508, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-5 x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = 6 a{\left(n \right)} - 3 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = 6 a{\left(n \right)} - 3 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(11 \,7^{\frac{2}{3}} \left(\sqrt{2}-\frac{9}{11}\right) \left(1+i \sqrt{3}\right) \left(9+4 \sqrt{2}\right)^{\frac{1}{3}}+98+9 \,7^{\frac{1}{3}} \left(i \sqrt{3}-1\right) \left(\sqrt{2}-\frac{1}{9}\right) \left(9+4 \sqrt{2}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\sqrt{2}-\frac{1}{2}\right) \left(1+i \sqrt{3}\right) \left(441+196 \sqrt{2}\right)^{\frac{1}{3}}}{42}+\frac{1}{6}+\frac{\left(\sqrt{2}-\frac{1}{2}\right) 7^{\frac{1}{3}} \left(i \sqrt{3}-1\right) \left(9+4 \sqrt{2}\right)^{\frac{2}{3}}}{42}\right)^{-n}}{1764}\\+\\\frac{\left(-11 \,7^{\frac{2}{3}} \left(\sqrt{2}-\frac{9}{11}\right) \left(i \sqrt{3}-1\right) \left(9+4 \sqrt{2}\right)^{\frac{1}{3}}+98-9 \left(1+i \sqrt{3}\right) 7^{\frac{1}{3}} \left(\sqrt{2}-\frac{1}{9}\right) \left(9+4 \sqrt{2}\right)^{\frac{2}{3}}\right) \left(-\frac{\left(\sqrt{2}-\frac{1}{2}\right) \left(i \sqrt{3}-1\right) \left(441+196 \sqrt{2}\right)^{\frac{1}{3}}}{42}+\frac{1}{6}-\frac{\left(\sqrt{2}-\frac{1}{2}\right) 7^{\frac{1}{3}} \left(1+i \sqrt{3}\right) \left(9+4 \sqrt{2}\right)^{\frac{2}{3}}}{42}\right)^{-n}}{1764}\\+\\\frac{\left(\left(-\frac{11 \,7^{\frac{2}{3}} \sqrt{2}}{9}+7^{\frac{2}{3}}\right) \left(9+4 \sqrt{2}\right)^{\frac{1}{3}}+\frac{49}{9}+7^{\frac{1}{3}} \left(\sqrt{2}-\frac{1}{9}\right) \left(9+4 \sqrt{2}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-2 \sqrt{2}+1\right) 7^{\frac{2}{3}} \left(1+2 \sqrt{2}\right)^{\frac{2}{3}}}{42}+\frac{\left(7+14 \sqrt{2}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}\right)^{-n}}{98} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 65 rules.
Finding the specification took 74 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_{18}\! \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_{32}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{18}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{18}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 0\\
F_{61}\! \left(x \right) &= F_{18}\! \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_{11}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{52}\! \left(x \right)\\
\end{align*}\)