Av(1243, 1342, 1423, 2134, 2314)
Generating Function
\(\displaystyle \frac{\left(x^{3}+x^{2}+x -1+\left(x^{3}+x^{2}-x +1\right) \sqrt{1-4 x}\right) \left(x -1\right)}{2 x \left(x^{6}+2 x^{5}-x^{4}+x^{3}+3 x^{2}-3 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 187, 610, 2037, 6927, 23910, 83578, 295309, 1053112, 3785718, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{6}+2 x^{5}-x^{4}+x^{3}+3 x^{2}-3 x +1\right) F \left(x
\right)^{2}-\left(x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 187\)
\(\displaystyle a \! \left(7\right) = 610\)
\(\displaystyle a \! \left(8\right) = 2037\)
\(\displaystyle a \! \left(9\right) = 6927\)
\(\displaystyle a \! \left(10\right) = 23910\)
\(\displaystyle a \! \left(n +11\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{12+n}-\frac{\left(16+7 n \right) a \! \left(1+n \right)}{12+n}+\frac{14 \left(n +5\right) a \! \left(n +2\right)}{12+n}+\frac{\left(n +6\right) a \! \left(n +3\right)}{12+n}-\frac{\left(184+33 n \right) a \! \left(n +4\right)}{12+n}+\frac{2 \left(145+22 n \right) a \! \left(n +5\right)}{12+n}-\frac{\left(36+n \right) a \! \left(n +6\right)}{12+n}-\frac{2 \left(165+23 n \right) a \! \left(n +7\right)}{12+n}+\frac{\left(478+55 n \right) a \! \left(n +8\right)}{12+n}-\frac{\left(306+31 n \right) a \! \left(n +9\right)}{12+n}+\frac{\left(98+9 n \right) a \! \left(n +10\right)}{12+n}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 187\)
\(\displaystyle a \! \left(7\right) = 610\)
\(\displaystyle a \! \left(8\right) = 2037\)
\(\displaystyle a \! \left(9\right) = 6927\)
\(\displaystyle a \! \left(10\right) = 23910\)
\(\displaystyle a \! \left(n +11\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{12+n}-\frac{\left(16+7 n \right) a \! \left(1+n \right)}{12+n}+\frac{14 \left(n +5\right) a \! \left(n +2\right)}{12+n}+\frac{\left(n +6\right) a \! \left(n +3\right)}{12+n}-\frac{\left(184+33 n \right) a \! \left(n +4\right)}{12+n}+\frac{2 \left(145+22 n \right) a \! \left(n +5\right)}{12+n}-\frac{\left(36+n \right) a \! \left(n +6\right)}{12+n}-\frac{2 \left(165+23 n \right) a \! \left(n +7\right)}{12+n}+\frac{\left(478+55 n \right) a \! \left(n +8\right)}{12+n}-\frac{\left(306+31 n \right) a \! \left(n +9\right)}{12+n}+\frac{\left(98+9 n \right) a \! \left(n +10\right)}{12+n}, \quad n \geq 11\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 36 rules.
Found on January 20, 2022.Finding the specification took 13 seconds.
Copy 36 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{22}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
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_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x , 1\right)\\
F_{28}\! \left(x , y\right) &= \frac{y F_{29}\! \left(x , y\right)-F_{29}\! \left(x , 1\right)}{-1+y}\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= y x\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{8}\! \left(x \right)\\
\end{align*}\)