Av(1243, 1342, 2134, 3142)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}{x^{6}-5 x^{5}+16 x^{4}-23 x^{3}+18 x^{2}-7 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 205, 638, 1979, 6138, 19043, 59086, 183323, 568751, 1764444, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-5 x^{5}+16 x^{4}-23 x^{3}+18 x^{2}-7 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+5 a \! \left(n +1\right)-16 a \! \left(n +2\right)+23 a \! \left(n +3\right)-18 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+5 a \! \left(n +1\right)-16 a \! \left(n +2\right)+23 a \! \left(n +3\right)-18 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{5395 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +4}}{18778}-\frac{5395 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +4}}{18778}+\frac{24215 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +3}}{18778}+\frac{24215 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +3}}{18778}-\frac{73523 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +2}}{18778}-\frac{73523 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +2}}{18778}+\frac{41439 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +1}}{9389}+\frac{41439 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +1}}{9389}+\frac{4652 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n -1}}{9389}+\frac{4652 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n -1}}{9389}-\frac{22178 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n}}{9389}-\frac{22178 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+16 Z^{4}-23 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n}}{9389}\)
This specification was found using the strategy pack "Point Placements" and has 57 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 57 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_{22}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \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_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{37}\! \left(x \right)\\
\end{align*}\)