Av(12345, 12354, 12435, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14325, 14352, 31245, 31254, 31425, 31452, 31524, 31542, 34125, 34152, 41235, 41325, 41352, 43125, 43152)
Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(2 x^{6}+12 x^{5}-19 x^{4}-2 x^{3}+14 x^{2}-7 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 95, 363, 1345, 4901, 17706, 63725, 229000, 822513, 2953951, 10608974, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{6}+12 x^{5}-19 x^{4}-2 x^{3}+14 x^{2}-7 x +1\right) F \! \left(x \right)-\left(x +1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 363\)
\(\displaystyle a{\left(n + 6 \right)} = - 2 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 19 a{\left(n + 2 \right)} + 2 a{\left(n + 3 \right)} - 14 a{\left(n + 4 \right)} + 7 a{\left(n + 5 \right)} - 2, \quad n \geq 7\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 363\)
\(\displaystyle a{\left(n + 6 \right)} = - 2 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 19 a{\left(n + 2 \right)} + 2 a{\left(n + 3 \right)} - 14 a{\left(n + 4 \right)} + 7 a{\left(n + 5 \right)} - 2, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -2-\frac{52596808 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +5}}{1363079}-\frac{52596808 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +5}}{1363079}-\frac{52596808 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +5}}{1363079}-\frac{52596808 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +5}}{1363079}-\frac{52596808 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +5}}{1363079}-\frac{52596808 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +5}}{1363079}-\frac{303947482 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +4}}{1363079}-\frac{303947482 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +4}}{1363079}-\frac{303947482 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +4}}{1363079}-\frac{303947482 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +4}}{1363079}-\frac{303947482 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +4}}{1363079}-\frac{303947482 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +4}}{1363079}+\frac{578169422 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +3}}{1363079}+\frac{578169422 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +3}}{1363079}+\frac{578169422 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +3}}{1363079}+\frac{578169422 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +3}}{1363079}+\frac{578169422 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +3}}{1363079}+\frac{578169422 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +3}}{1363079}+\frac{37722 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +2}}{71741}+\frac{37722 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +2}}{71741}+\frac{37722 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +2}}{71741}+\frac{37722 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +2}}{71741}+\frac{37722 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +2}}{71741}+\frac{37722 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +2}}{71741}-\frac{419081811 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +1}}{1363079}-\frac{419081811 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +1}}{1363079}-\frac{419081811 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +1}}{1363079}-\frac{419081811 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +1}}{1363079}-\frac{419081811 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +1}}{1363079}-\frac{419081811 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +1}}{1363079}-\frac{33841512 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n -1}}{1363079}-\frac{33841512 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n -1}}{1363079}-\frac{33841512 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n -1}}{1363079}-\frac{33841512 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n -1}}{1363079}-\frac{33841512 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n -1}}{1363079}-\frac{33841512 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n -1}}{1363079}+\frac{11992385 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n}}{71741}+\frac{11992385 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n}}{71741}+\frac{11992385 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n}}{71741}+\frac{11992385 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n}}{71741}+\frac{11992385 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n}}{71741}+\frac{11992385 \mathit{RootOf} \left(2 Z^{6}+12 Z^{5}-19 Z^{4}-2 Z^{3}+14 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n}}{71741}\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 103 rules.
Finding the specification took 85 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_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{19}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{35}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{19}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{19}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{19}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{19}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{51}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{19}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{19}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= x^{2}\\
F_{61}\! \left(x \right) &= F_{19}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{19}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{19}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 0\\
F_{68}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{51}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 0\\
F_{71}\! \left(x \right) &= F_{19}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{73}\! \left(x \right) &= 0\\
F_{74}\! \left(x \right) &= F_{19}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{19}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{79}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{19}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{88}\! \left(x \right)+F_{95}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{19}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{88}\! \left(x \right)+F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{19}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 0\\
F_{94}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{19}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{97}\! \left(x \right) &= 0\\
F_{98}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{71}\! \left(x \right)+F_{97}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{69}\! \left(x \right)+F_{93}\! \left(x \right)+F_{99}\! \left(x \right)\\
\end{align*}\)