Av(1243, 1342, 1423, 2413, 3214)
Generating Function
\(\displaystyle \frac{\left(x -1\right)^{3}}{x^{7}-2 x^{6}-x^{5}+x^{4}+4 x^{3}-5 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 144, 385, 1046, 2860, 7808, 21268, 57899, 157676, 429522, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-2 x^{6}-x^{5}+x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 144\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)-5 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 144\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)-5 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +5}}{53464773}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +4}}{17821591}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +3}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +2}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n -1}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf} \left(Z^{7}-2 Z^{6}-Z^{5}+Z^{4}+4 Z^{3}-5 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n}}{53464773}\)
This specification was found using the strategy pack "Point Placements" and has 79 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
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\(\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_{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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{52}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{68}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{50}\! \left(x \right)\\
\end{align*}\)