Av(1234, 1342, 1432, 2413)
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Generating Function
\(\displaystyle -\frac{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}{\left(x^{2}-x +1\right) \left(x^{6}-x^{5}-2 x^{4}+7 x^{3}-11 x^{2}+6 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 201, 629, 1968, 6155, 19239, 60109, 187745, 586295, 1830674, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-x +1\right) \left(x^{6}-x^{5}-2 x^{4}+7 x^{3}-11 x^{2}+6 x -1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(7\right) = 629\)
\(\displaystyle a \! \left(n +1\right) = \frac{a \! \left(n \right)}{2}+4 a \! \left(n +3\right)-10 a \! \left(n +4\right)+12 a \! \left(n +5\right)-9 a \! \left(n +6\right)+\frac{7 a \! \left(n +7\right)}{2}-\frac{a \! \left(n +8\right)}{2}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{713439 \,\mathrm{I}}{5268893} \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)-1\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)-3\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)+2\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}-\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)+2\right) \left(-\frac{1395125 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{713439}+\frac{1423574 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\frac{1423574 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\frac{1423574 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\frac{586438 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}-\frac{117970 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}-\frac{117970 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}-\frac{117970 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}-\frac{117970 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\frac{1423574 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\frac{1423574 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}-\frac{617378 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)}{79271}-\frac{617378 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{79271}-\frac{617378 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{79271}-\frac{617378 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{79271}-\frac{617378 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{79271}-\frac{617378 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{79271}-\frac{117970 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)}{237813}-\frac{117970 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}-\frac{1395125 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)}{713439}-\frac{1395125 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{713439}-\frac{1395125 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{713439}-\frac{1395125 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{713439}-\frac{1395125 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{713439}+\frac{1423574 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)}{237813}+\left(\frac{586438 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \left(\frac{586438 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \left(\frac{586438 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \left(\frac{586438 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)}{237813}+\left(\frac{586438 \,\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{237813}+\left(\left(\frac{752699 \sqrt{3}}{4280634}+\frac{752699 \,\mathrm{I}}{475626}\right) \left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}+\frac{752699 \left(\frac{1}{2}+\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n} \left(\mathrm{I}-\frac{\sqrt{3}}{9}\right)}{475626}\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)+\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +5}\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)\right)\right)\right)\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)+\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +5} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)+\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +5} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)+\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +5} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)+\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +5} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)+\mathrm{I} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +5} \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{2}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)-3\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{2}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)-3\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)-1\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{2}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)-3\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}-\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)+2\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)-1\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =1\right)-3\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)-1\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =3\right)-1\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)^{2}-\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =6\right)+2\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)^{2}-\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =5\right)+2\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)-1\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =2\right)-3\right) \left(\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)^{2}-\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}+7 Z^{3}-11 Z^{2}+6 Z -1, \mathit{index} =4\right)+2\right)\)
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This specification was found using the strategy pack "Point Placements" and has 151 rules.

Found on January 18, 2022.

Finding the specification took 7 seconds.

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\(\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_{148}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \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_{39}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \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_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{17}\! \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_{30}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{145}\! \left(x \right)+F_{22}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 0\\ F_{52}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{144}\! \left(x \right)+F_{22}\! \left(x \right)+F_{54}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{22}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{99}\! \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_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 0\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{74}\! \left(x \right)+F_{95}\! \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_{78}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{80}\! \left(x \right)+F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{86}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{90}\! \left(x \right) &= 0\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{94}\! \left(x \right) &= 0\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{99}\! \left(x \right) &= 0\\ F_{100}\! \left(x \right) &= 0\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{143}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{135}\! \left(x \right)+F_{138}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{120}\! \left(x \right) &= 0\\ F_{121}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{123}\! \left(x \right)+F_{134}\! \left(x \right)+F_{22}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{133}\! \left(x \right)+F_{22}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{126}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{133}\! \left(x \right) &= 0\\ F_{134}\! \left(x \right) &= 0\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{144}\! \left(x \right) &= 0\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{2}\! \left(x \right)\\ \end{align*}\)