Av(1342, 1423, 3214)
View Raw Data
Generating Function
\(\displaystyle -\frac{\left(x^{2}+x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}{x^{9}-2 x^{8}+6 x^{7}-4 x^{6}-7 x^{5}+32 x^{4}-40 x^{3}+25 x^{2}-8 x +1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 21, 73, 240, 759, 2365, 7369, 23069, 72495, 228186, 718341, 2260566, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{9}-2 x^{8}+6 x^{7}-4 x^{6}-7 x^{5}+32 x^{4}-40 x^{3}+25 x^{2}-8 x +1\right) F \! \left(x \right)+\left(x^{2}+x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 240\)
\(\displaystyle a \! \left(7\right) = 759\)
\(\displaystyle a \! \left(8\right) = 2365\)
\(\displaystyle a \! \left(n +9\right) = -a \! \left(n \right)+2 a \! \left(n +1\right)-6 a \! \left(n +2\right)+4 a \! \left(n +3\right)+7 a \! \left(n +4\right)-32 a \! \left(n +5\right)+40 a \! \left(n +6\right)-25 a \! \left(n +7\right)+8 a \! \left(n +8\right), \quad n \geq 9\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle -\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +7}}{8887218836975}-\frac{1564927109193 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +7}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +6}}{8887218836975}+\frac{2319611559304 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +6}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +5}}{8887218836975}-\frac{8158171237737 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +5}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +4}}{8887218836975}+\frac{2082480184784 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +4}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +3}}{8887218836975}+\frac{12235877201842 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +3}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +2}}{8887218836975}-\frac{43703293918168 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +2}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +1}}{8887218836975}+\frac{40041349058213 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n +1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n -1}}{8887218836975}+\frac{3501868394607 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n -1}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n}}{8887218836975}-\frac{17558611111888 \mathit{RootOf} \left(Z^{9}-2 Z^{8}+6 Z^{7}-4 Z^{6}-7 Z^{5}+32 Z^{4}-40 Z^{3}+25 Z^{2}-8 Z +1, \mathit{index} =9\right)^{-n}}{8887218836975}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 84 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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

Copy 84 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{51}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{67}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{80}\! \left(x \right)\\ \end{align*}\)