Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 21354, 21453, 21543, 23451, 23541, 24351, 31452, 31542, 32451, 32541, 41532, 42531)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right)}{\left(x^{2}+x -1\right) \left(2 x^{5}-12 x^{4}+26 x^{3}-22 x^{2}+8 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 408, 1620, 6310, 24298, 92992, 354822, 1352030, 5149008, 19605494, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(2 x^{5}-12 x^{4}+26 x^{3}-22 x^{2}+8 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}-3 x +1\right) \left(2 x^{3}+2 x^{2}-4 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 408\)
\(\displaystyle a{\left(n + 7 \right)} = - 2 a{\left(n \right)} + 10 a{\left(n + 1 \right)} - 12 a{\left(n + 2 \right)} - 16 a{\left(n + 3 \right)} + 40 a{\left(n + 4 \right)} - 29 a{\left(n + 5 \right)} + 9 a{\left(n + 6 \right)}, \quad n \geq 7\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 408\)
\(\displaystyle a{\left(n + 7 \right)} = - 2 a{\left(n \right)} + 10 a{\left(n + 1 \right)} - 12 a{\left(n + 2 \right)} - 16 a{\left(n + 3 \right)} + 40 a{\left(n + 4 \right)} - 29 a{\left(n + 5 \right)} + 9 a{\left(n + 6 \right)}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{376384 \left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{4}-6 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{3}+13 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{2}-11 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)+4\right) \left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{4}-6 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{3}+13 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{2}-11 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)+4\right) \left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{4}-6 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{3}+13 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{2}-11 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)+4\right) \left(-\frac{74861 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{11762}+\frac{123493 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{11762}-\frac{32697 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{5881}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)-\frac{74861 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}+\frac{123493 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}-\frac{32697 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{5881}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)-\frac{74861 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}+\frac{123493 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}-\frac{32697 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{5881}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)-\frac{74861 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}+\frac{123493 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}-\frac{32697 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{5881}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)-\frac{74861 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +1} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}+\frac{123493 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{11762}-\frac{32697 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)}{5881}+\frac{11725 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{11762}+\left(\frac{11725 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{11762}+\left(\frac{11725 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{11762}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right) \left(\frac{11725 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n} \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)}{11762}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right) \left(\frac{11725 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n}}{11762}+\left(\left(\frac{1383}{11762}-\frac{5993 \sqrt{5}}{58810}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+\frac{5993 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n} \left(\sqrt{5}+\frac{15}{13}\right)}{58810}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +4}\right)\right)\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)\right) \left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{4}-6 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{3}+13 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{2}-11 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)+4\right) \left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{4}-6 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{3}+13 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{2}-11 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+26 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)+4\right)}{14291}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 101 rules.
Finding the specification took 103 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_{18}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{8}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{47}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{18}\! \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_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{29}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{18}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{53}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{53}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)+F_{7}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{18}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 0\\
F_{65}\! \left(x \right) &= F_{18}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{41}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{18}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{7}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{18}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{7}\! \left(x \right)+F_{70}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{18}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{28}\! \left(x \right)+F_{65}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{18}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{7}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{18}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{82}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{18}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{70}\! \left(x \right)+F_{93}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{18}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{18}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{7}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{11}\! \left(x \right)\\
\end{align*}\)