Av(1234, 1243, 2314, 3214)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{2}+x +1\right) \left(x^{3}+2 x^{2}+2 x -1\right)}{x^{6}+2 x^{5}+3 x^{4}-x^{3}-x^{2}-3 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 213, 701, 2307, 7594, 25001, 82310, 270989, 892181, 2937344, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}+2 x^{5}+3 x^{4}-x^{3}-x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}+x +1\right) \left(x^{3}+2 x^{2}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 213\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 213\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{55 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +4}}{3471}-\frac{55 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +4}}{3471}+\frac{62 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +3}}{10413}+\frac{62 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +3}}{10413}+\frac{12 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +2}}{1157}+\frac{12 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +2}}{1157}+\frac{1402 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +1}}{10413}+\frac{1402 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +1}}{10413}+\frac{512 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n -1}}{10413}+\frac{512 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n -1}}{10413}-\frac{382 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{10413}-\frac{382 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{10413}-\frac{382 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{10413}-\frac{382 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{10413}-\frac{382 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{10413}-\frac{382 \mathit{RootOf} \left(Z^{6}+2 Z^{5}+3 Z^{4}-Z^{3}-Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n}}{10413}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 70 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
Copy 70 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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \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) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \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_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \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_{41}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 3 F_{13}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \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_{54}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
\end{align*}\)