Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13452, 13542, 14253, 14352, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24153, 31245, 31254, 31425, 31524, 32145, 32154, 41235, 41325, 42135)
View Raw Data
Generating Function
\(\displaystyle -\frac{x^{9}-5 x^{7}+2 x^{6}+5 x^{5}-3 x^{4}-3 x^{3}+2 x^{2}+3 x -1}{x^{10}-x^{9}-4 x^{8}+7 x^{7}+5 x^{6}-7 x^{5}-2 x^{4}+5 x^{3}-4 x +1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 24, 90, 334, 1235, 4567, 16906, 62593, 231752, 858057, 3176909, 11762326, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{10}-x^{9}-4 x^{8}+7 x^{7}+5 x^{6}-7 x^{5}-2 x^{4}+5 x^{3}-4 x +1\right) F \! \left(x \right)+x^{9}-5 x^{7}+2 x^{6}+5 x^{5}-3 x^{4}-3 x^{3}+2 x^{2}+3 x -1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 334\)
\(\displaystyle a(7) = 1235\)
\(\displaystyle a(8) = 4567\)
\(\displaystyle a(9) = 16906\)
\(\displaystyle a{\left(n + 7 \right)} = - \frac{a{\left(n \right)}}{5} + \frac{a{\left(n + 1 \right)}}{5} + \frac{4 a{\left(n + 2 \right)}}{5} - \frac{7 a{\left(n + 3 \right)}}{5} - a{\left(n + 4 \right)} + \frac{7 a{\left(n + 5 \right)}}{5} + \frac{2 a{\left(n + 6 \right)}}{5} + \frac{4 a{\left(n + 9 \right)}}{5} - \frac{a{\left(n + 10 \right)}}{5}, \quad n \geq 10\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +8}}{23555872254511183}+\frac{191779882428321 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +8}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +7}}{23555872254511183}-\frac{255858936414111 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +7}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +6}}{23555872254511183}-\frac{845620074863939 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +6}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +5}}{23555872254511183}+\frac{1775022649729563 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +5}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +4}}{23555872254511183}+\frac{959364711490387 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +4}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +3}}{23555872254511183}-\frac{2969547211032564 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +3}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +2}}{23555872254511183}-\frac{90619013943461 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +2}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n +1}}{23555872254511183}+\frac{2757460152444439 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n +1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n -1}}{23555872254511183}+\frac{195273178122854 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n -1}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =1\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =2\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =3\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =4\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =5\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =6\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =7\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =8\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =9\right)^{-n}}{23555872254511183}+\frac{1636199933332839 \mathit{RootOf} \left(Z^{10}-Z^{9}-4 Z^{8}+7 Z^{7}+5 Z^{6}-7 Z^{5}-2 Z^{4}+5 Z^{3}-4 Z +1, \mathit{index} =10\right)^{-n}}{23555872254511183}\)
Copy to clipboard:

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 137 rules.

Finding the specification took 173 seconds.

This tree is too big to show here. Click to view tree on new page.

Copy 137 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_{14}\! \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_{40}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{14}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \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_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= x^{2}\\ F_{36}\! \left(x \right) &= x^{2}\\ F_{37}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{24}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{24}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{35}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{14}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 0\\ F_{65}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{14}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{77}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{14}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{14}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 3 F_{24}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{14}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{14}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{84}\! \left(x \right) &= 0\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{78}\! \left(x \right)+F_{87}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{14}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{90}\! \left(x \right) &= 0\\ F_{91}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{81}\! \left(x \right)+F_{92}\! \left(x \right)+F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{14}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{95}\! \left(x \right) &= 0\\ F_{96}\! \left(x \right) &= 0\\ F_{97}\! \left(x \right) &= F_{14}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{102}\! \left(x \right) &= 0\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{114}\! \left(x \right)+F_{117}\! \left(x \right)+F_{118}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{105}\! \left(x \right) &= 2 F_{24}\! \left(x \right)+F_{106}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{117}\! \left(x \right) &= 0\\ F_{118}\! \left(x \right) &= 0\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{24}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{24}\! \left(x \right)+F_{67}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{24}\! \left(x \right)+F_{71}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{82}\! \left(x \right)\\ \end{align*}\)