Av(1243, 1342, 1432, 2143, 2314)
Generating Function
\(\displaystyle -\frac{\left(-1+2 x \right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}{x^{7}+2 x^{6}-5 x^{5}+x^{4}+9 x^{3}-12 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 173, 512, 1512, 4462, 13163, 38824, 114502, 337690, 995921, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+2 x^{6}-5 x^{5}+x^{4}+9 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+\left(-1+2 x \right) \left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 173\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-5 a \! \left(n +2\right)+a \! \left(n +3\right)+9 a \! \left(n +4\right)-12 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 173\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-5 a \! \left(n +2\right)+a \! \left(n +3\right)+9 a \! \left(n +4\right)-12 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +5}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +4}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +3}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +2}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n -1}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf} \left(Z^{7}+2 Z^{6}-5 Z^{5}+Z^{4}+9 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n}}{9050009}\)
This specification was found using the strategy pack "Point Placements" and has 52 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 52 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{18}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= x^{2}\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 0\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
\end{align*}\)