Av(1243, 1342, 2413, 3124)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}{3 x^{7}-16 x^{6}+48 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 212, 669, 2094, 6535, 20376, 63513, 197946, 616879, 1922397, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{7}-16 x^{6}+48 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 212\)
\(\displaystyle a \! \left(n +7\right) = 3 a \! \left(n \right)-16 a \! \left(n +1\right)+48 a \! \left(n +2\right)-74 a \! \left(n +3\right)+65 a \! \left(n +4\right)-33 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 212\)
\(\displaystyle a \! \left(n +7\right) = 3 a \! \left(n \right)-16 a \! \left(n +1\right)+48 a \! \left(n +2\right)-74 a \! \left(n +3\right)+65 a \! \left(n +4\right)-33 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +5}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +4}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +3}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +2}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n -1}}{7727983}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf} \left(3 Z^{7}-16 Z^{6}+48 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n}}{145811}\)
This specification was found using the strategy pack "Point Placements" and has 38 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 38 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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{0}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{35}\! \left(x \right)\\
\end{align*}\)