Av(1243, 2143, 2413, 3142, 3214)
Generating Function
\(\displaystyle -\frac{\left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3}}{x^{7}-12 x^{6}+34 x^{5}-52 x^{4}+47 x^{3}-26 x^{2}+8 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 173, 514, 1533, 4591, 13782, 41418, 124531, 374529, 1126620, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-12 x^{6}+34 x^{5}-52 x^{4}+47 x^{3}-26 x^{2}+8 x -1\right) F \! \left(x \right)+\left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 173\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-12 a \! \left(n +1\right)+34 a \! \left(n +2\right)-52 a \! \left(n +3\right)+47 a \! \left(n +4\right)-26 a \! \left(n +5\right)+8 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 173\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-12 a \! \left(n +1\right)+34 a \! \left(n +2\right)-52 a \! \left(n +3\right)+47 a \! \left(n +4\right)-26 a \! \left(n +5\right)+8 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +5}}{128257}+\frac{17672 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +5}}{128257}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +4}}{384771}-\frac{612028 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +4}}{384771}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +3}}{128257}+\frac{505973 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +3}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +2}}{128257}-\frac{666681 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +2}}{128257}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +1}}{384771}+\frac{1439041 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n -1}}{384771}+\frac{113239 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n -1}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n}}{384771}-\frac{559823 \mathit{RootOf} \left(Z^{7}-12 Z^{6}+34 Z^{5}-52 Z^{4}+47 Z^{3}-26 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n}}{384771}\)
This specification was found using the strategy pack "Point Placements" and has 71 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 71 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_{28}\! \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_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \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_{18}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \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_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \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_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
\end{align*}\)