Av(1243, 2143, 2314, 2413, 3142)
Generating Function
\(\displaystyle \frac{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}{\left(x -1\right) \left(x^{5}-3 x^{4}+13 x^{3}-16 x^{2}+7 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 188, 586, 1822, 5660, 17581, 54622, 169757, 527741, 1641077, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{5}-3 x^{4}+13 x^{3}-16 x^{2}+7 x -1\right) F \! \left(x \right)-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+13 a \! \left(n +2\right)-16 a \! \left(n +3\right)+7 a \! \left(n +4\right)-1, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+13 a \! \left(n +2\right)-16 a \! \left(n +3\right)+7 a \! \left(n +4\right)-1, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle 1+\frac{10759 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +4}}{4903}+\frac{10759 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +4}}{4903}+\frac{10759 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +4}}{4903}+\frac{10759 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +4}}{4903}+\frac{10759 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +4}}{4903}-\frac{33483 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{4903}-\frac{33483 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{4903}-\frac{33483 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{4903}-\frac{33483 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{4903}-\frac{33483 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{4903}+\frac{142236 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{4903}+\frac{142236 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{4903}+\frac{142236 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{4903}+\frac{142236 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{4903}+\frac{142236 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{4903}-\frac{184901 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{4903}-\frac{184901 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{4903}-\frac{184901 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{4903}-\frac{184901 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{4903}-\frac{184901 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{4903}-\frac{11158 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{4903}-\frac{11158 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{4903}-\frac{11158 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{4903}-\frac{11158 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{4903}-\frac{11158 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{4903}+\frac{81450 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{4903}+\frac{81450 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{4903}+\frac{81450 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{4903}+\frac{81450 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{4903}+\frac{81450 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+13 Z^{3}-16 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{4903}\)
This specification was found using the strategy pack "Point Placements" and has 52 rules.
Found on January 18, 2022.Finding the specification took 0 seconds.
Copy 52 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_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \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_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \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) &= F_{10}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
\end{align*}\)