Av(1234, 1243, 1432, 2314, 3214)
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Generating Function
\(\displaystyle -\frac{2 x^{3}+x^{2}-1}{4 x^{9}+10 x^{8}+8 x^{7}-2 x^{6}-8 x^{5}-5 x^{4}-4 x^{3}-2 x^{2}-x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 52, 134, 354, 959, 2597, 6983, 18739, 50358, 135465, 364366, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{9}+10 x^{8}+8 x^{7}-2 x^{6}-8 x^{5}-5 x^{4}-4 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+2 x^{3}+x^{2}-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 134\)
\(\displaystyle a \! \left(7\right) = 354\)
\(\displaystyle a \! \left(8\right) = 959\)
\(\displaystyle a \! \left(n +9\right) = -4 a \! \left(n \right)-10 a \! \left(n +1\right)-8 a \! \left(n +2\right)+2 a \! \left(n +3\right)+8 a \! \left(n +4\right)+5 a \! \left(n +5\right)+4 a \! \left(n +6\right)+2 a \! \left(n +7\right)+a \! \left(n +8\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle -\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +7}}{5725542899}-\frac{531374572 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +7}}{5725542899}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +6}}{40078800293}-\frac{6880792614 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +6}}{40078800293}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +5}}{5725542899}-\frac{475066916 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +5}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +4}}{5725542899}-\frac{58665571 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +4}}{5725542899}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +3}}{40078800293}-\frac{805421045 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +3}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +2}}{40078800293}+\frac{785664660 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +2}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +1}}{40078800293}+\frac{6128277060 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n -1}}{40078800293}+\frac{2115360020 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n -1}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n}}{40078800293}+\frac{6152067710 \mathit{RootOf} \left(4 Z^{9}+10 Z^{8}+8 Z^{7}-2 Z^{6}-8 Z^{5}-5 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n}}{40078800293}\)
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This specification was found using the strategy pack "Point Placements" and has 76 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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