Av(1342, 1423, 1432, 2143, 2314)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(2 x -1\right) \left(x^{2}+x -1\right)}{3 x^{5}-3 x^{4}-2 x^{3}+7 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 167, 486, 1414, 4116, 11984, 34893, 101593, 295788, 861179, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{5}-3 x^{4}-2 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x -1\right) \left(x^{2}+x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -3 a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = -3 a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{11418 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}}{23981}-\frac{11418 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}}{23981}-\frac{11418 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +3}}{23981}-\frac{11418 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +3}}{23981}-\frac{11418 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +3}}{23981}+\frac{225 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}}{23981}+\frac{225 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}}{23981}+\frac{225 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}}{23981}+\frac{225 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +2}}{23981}+\frac{225 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +2}}{23981}+\frac{10334 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}}{23981}+\frac{10334 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}}{23981}+\frac{10334 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}}{23981}+\frac{10334 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}}{23981}+\frac{10334 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +1}}{23981}+\frac{5213 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n -1}}{23981}+\frac{5213 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n -1}}{23981}+\frac{5213 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n -1}}{23981}+\frac{5213 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n -1}}{23981}+\frac{5213 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n -1}}{23981}-\frac{11723 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}}{23981}-\frac{11723 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}}{23981}-\frac{11723 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}}{23981}-\frac{11723 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}}{23981}-\frac{11723 \mathit{RootOf} \left(3 Z^{5}-3 Z^{4}-2 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{23981}\)
This specification was found using the strategy pack "Point Placements" and has 47 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 47 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{10}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= x^{2}\\
F_{38}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\
\end{align*}\)