Av(1234, 1342, 1432, 3124, 3214)
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Generating Function
\(\displaystyle -\frac{x^{3}+x^{2}-1}{x^{9}+5 x^{8}+7 x^{7}+x^{6}-8 x^{5}-6 x^{4}-3 x^{3}-2 x^{2}-x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 51, 126, 329, 882, 2350, 6193, 16305, 43081, 113941, 301110, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{9}+5 x^{8}+7 x^{7}+x^{6}-8 x^{5}-6 x^{4}-3 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+x^{3}+x^{2}-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 126\)
\(\displaystyle a \! \left(7\right) = 329\)
\(\displaystyle a \! \left(8\right) = 882\)
\(\displaystyle a \! \left(n +9\right) = -a \! \left(n \right)-5 a \! \left(n +1\right)-7 a \! \left(n +2\right)-a \! \left(n +3\right)+8 a \! \left(n +4\right)+6 a \! \left(n +5\right)+3 a \! \left(n +6\right)+2 a \! \left(n +7\right)+a \! \left(n +8\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle -\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +7}}{18177039023807}-\frac{70147941540 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +7}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +6}}{18177039023807}-\frac{616666782609 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +6}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +5}}{18177039023807}-\frac{1721658224701 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +5}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +4}}{18177039023807}-\frac{1804960996571 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +4}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +3}}{18177039023807}-\frac{676125178963 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +3}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +2}}{18177039023807}+\frac{955319857718 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +2}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +1}}{18177039023807}+\frac{1947700070921 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n -1}}{18177039023807}+\frac{892324086899 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n -1}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n}}{18177039023807}+\frac{3634472439993 \mathit{RootOf} \left(Z^{9}+5 Z^{8}+7 Z^{7}+Z^{6}-8 Z^{5}-6 Z^{4}-3 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n}}{18177039023807}\)
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This specification was found using the strategy pack "Point Placements" and has 87 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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Copy 87 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_{20}\! \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_{14}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{38}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{74}\! \left(x \right)+F_{82}\! \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_{79}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{11}\! \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_{54}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{34}\! \left(x \right)\\ \end{align*}\)