Av(13425, 13452, 14235, 14253, 14325, 14352, 14523, 14532, 31425, 31452, 34125, 34152, 41235, 41253, 41325, 41352, 41523, 41532, 43125, 43152)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(5 x^{2}-5 x +1\right) \left(x -1\right)^{5}}{2 x^{9}+28 x^{8}-71 x^{7}+84 x^{6}-30 x^{5}-49 x^{4}+69 x^{3}-38 x^{2}+10 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 412, 1660, 6608, 26200, 103896, 412480, 1639152, 6517100, 25916388, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{9}+28 x^{8}-71 x^{7}+84 x^{6}-30 x^{5}-49 x^{4}+69 x^{3}-38 x^{2}+10 x -1\right) F \! \left(x \right)-\left(x +1\right) \left(5 x^{2}-5 x +1\right) \left(x -1\right)^{5} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 412\)
\(\displaystyle a(7) = 1660\)
\(\displaystyle a(8) = 6608\)
\(\displaystyle a{\left(n + 9 \right)} = 2 a{\left(n \right)} + 28 a{\left(n + 1 \right)} - 71 a{\left(n + 2 \right)} + 84 a{\left(n + 3 \right)} - 30 a{\left(n + 4 \right)} - 49 a{\left(n + 5 \right)} + 69 a{\left(n + 6 \right)} - 38 a{\left(n + 7 \right)} + 10 a{\left(n + 8 \right)}, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +7}}{1412661847026189}+\frac{44081793462596 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +7}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +6}}{1412661847026189}+\frac{614407746074546 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +6}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +5}}{1412661847026189}-\frac{1638347611117250 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +5}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +4}}{1412661847026189}+\frac{1413068713223275 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +4}}{1412661847026189}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +3}}{470887282342063}-\frac{61785192929775 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +3}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +2}}{470887282342063}-\frac{527334470355054 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +2}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +1}}{470887282342063}+\frac{407862150295021 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n +1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n -1}}{470887282342063}+\frac{9919119672148 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n -1}}{470887282342063}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n}}{1412661847026189}-\frac{179266696585979 \mathit{RootOf} \left(2 Z^{9}+28 Z^{8}-71 Z^{7}+84 Z^{6}-30 Z^{5}-49 Z^{4}+69 Z^{3}-38 Z^{2}+10 Z -1, \mathit{index} =9\right)^{-n}}{1412661847026189}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 88 rules.

Finding the specification took 86 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_{16}\! \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_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{16}\! \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_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \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_{40}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{16}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{54}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= x^{2}\\ F_{63}\! \left(x \right) &= F_{16}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{16}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{16}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{16}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{41}\! \left(x \right)+F_{54}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{16}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{16}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{16}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{16}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{53}\! \left(x \right)\\ \end{align*}\)