Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14325, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 24315, 31425, 31452, 31524, 31542, 32415, 32514, 41523, 41532, 42513)
View Raw Data
Generating Function
\(\displaystyle -\frac{\left(x^{4}+3 x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2}}{3 x^{9}+9 x^{8}+5 x^{7}-8 x^{6}-7 x^{5}-x^{4}+2 x^{3}+2 x^{2}-4 x +1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 24, 90, 311, 1034, 3401, 11198, 37004, 122569, 406315, 1346954, 4464407, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{9}+9 x^{8}+5 x^{7}-8 x^{6}-7 x^{5}-x^{4}+2 x^{3}+2 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x^{4}+3 x^{3}+x^{2}+x -1\right) \left(x^{2}+x -1\right)^{2} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 311\)
\(\displaystyle a(7) = 1034\)
\(\displaystyle a(8) = 3401\)
\(\displaystyle a{\left(n + 9 \right)} = - 3 a{\left(n \right)} - 9 a{\left(n + 1 \right)} - 5 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)} + 7 a{\left(n + 4 \right)} + a{\left(n + 5 \right)} - 2 a{\left(n + 6 \right)} - 2 a{\left(n + 7 \right)} + 4 a{\left(n + 8 \right)}, \quad n \geq 9\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +7}}{130471038855829}+\frac{8957293477068 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +7}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +6}}{130471038855829}+\frac{33517311326592 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +6}}{130471038855829}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +5}}{18638719836547}+\frac{4844802016148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +5}}{18638719836547}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +4}}{130471038855829}-\frac{17639193747478 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +4}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +3}}{130471038855829}-\frac{45558929917148 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +3}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +2}}{130471038855829}-\frac{22475478846936 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +2}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +1}}{130471038855829}+\frac{4629554314341 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n -1}}{130471038855829}+\frac{2709260063488 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n -1}}{130471038855829}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n}}{18638719836547}+\frac{3314052006278 \mathit{RootOf} \left(3 Z^{9}+9 Z^{8}+5 Z^{7}-8 Z^{6}-7 Z^{5}-Z^{4}+2 Z^{3}+2 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n}}{18638719836547}\)
Copy to clipboard:

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 126 rules.

Finding the specification took 118 seconds.

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

Copy 126 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_{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_{27}\! \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_{19}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \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_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{32}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{36}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{16}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{46}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{16}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{43}\! \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_{11}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{16}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{44}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{16}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{16}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{20}\! \left(x \right)+F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{16}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{20}\! \left(x \right)+F_{93}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{16}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{105}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{106}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{112}\! \left(x \right)+F_{20}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 0\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{118}\! \left(x \right)+F_{20}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{118}\! \left(x \right)+F_{20}\! \left(x \right)+F_{94}\! \left(x \right)\\ \end{align*}\)