Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13452, 13542, 14253, 14352, 21345, 21354, 21435, 21453, 21534, 21543, 23451, 23541, 24351, 31425, 31452, 31524, 31542, 32451, 32541, 41523, 41532, 42531)
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Generating Function
\(\displaystyle \frac{x^{10}+3 x^{9}-x^{8}-7 x^{7}-2 x^{6}-2 x^{5}-4 x^{4}+7 x^{3}+5 x^{2}-5 x +1}{\left(x^{5}+x^{4}-2 x^{3}+3 x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 24, 90, 312, 1029, 3304, 10431, 32530, 100452, 307656, 935740, 2829232, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+x^{4}-2 x^{3}+3 x -1\right)^{2} F \! \left(x \right)-x^{10}-3 x^{9}+x^{8}+7 x^{7}+2 x^{6}+2 x^{5}+4 x^{4}-7 x^{3}-5 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 312\)
\(\displaystyle a(7) = 1029\)
\(\displaystyle a(8) = 3304\)
\(\displaystyle a(9) = 10431\)
\(\displaystyle a(10) = 32530\)
\(\displaystyle a{\left(n + 10 \right)} = - a{\left(n \right)} - 2 a{\left(n + 1 \right)} + 3 a{\left(n + 2 \right)} + 4 a{\left(n + 3 \right)} - 10 a{\left(n + 4 \right)} - 4 a{\left(n + 5 \right)} + 14 a{\left(n + 6 \right)} - 4 a{\left(n + 7 \right)} - 9 a{\left(n + 8 \right)} + 6 a{\left(n + 9 \right)}, \quad n \geq 11\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}-\frac{36 \left(\left(\left(\left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-5\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-5 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-7\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\left(-5 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-7\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-7 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-13\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)+\left(\left(-5 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-7\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-7 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-13\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\left(-7 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-13\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-13 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-19\right) \left(\left(\left(\left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{29}{18}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{18}-\frac{1}{4}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\left(-\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{18}-\frac{1}{4}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-\frac{\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{4}+\frac{29}{72}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)+\left(\left(-\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{18}-\frac{1}{4}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)-\frac{\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{4}+\frac{29}{72}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\left(-\frac{\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{4}+\frac{29}{72}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)}{72}-\frac{113}{36}\right)}{461} & n =0 \\ \frac{19512 \left(\left(\left(\left(\frac{294 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{271}+\frac{762}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{1290}{271}+\frac{762 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(\frac{1290}{271}+\frac{762 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{2037}{271}+\frac{1290 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{49 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}+\frac{1717}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{2551}{1084}+\frac{1717 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(\frac{2551}{1084}+\frac{1717 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{4501}{1084}+\frac{2551 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{421 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{542}-\frac{903}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{1565}{542}-\frac{903 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{1565}{542}-\frac{903 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{2423}{542}-\frac{1565 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{271}-1\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{437}{271}-\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{437}{271}-\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{719}{271}-\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{248 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}+\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}}{271}+\frac{332 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}+\frac{149 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{762}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{1290}{271}-\frac{112 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}+\frac{708 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{105 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(\frac{1290}{271}-\frac{112 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}+\frac{708 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{105 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{155 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{148 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}}{271}+\frac{1156 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}+\frac{2037}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{332 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{547 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{451}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{945}{1084}+\frac{1297 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}+\frac{708 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{945}{1084}+\frac{1297 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}+\frac{708 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{1156 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{1931 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{1251}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(\frac{243 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}+\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{361}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{691}{542}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}+\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{691}{542}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}+\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{747 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{985}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{361}{542}-\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{243 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{526}{271}-\frac{395 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}-\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{907 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1493}{542}-\frac{243 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}-\frac{395 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}+\frac{529 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\frac{1493 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}+\frac{361 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}+\frac{526 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{599}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\left(\frac{691}{542}-\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}-\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{838}{271}-\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}-\frac{1929 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}-\frac{569 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{1731 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}-\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}-\frac{569 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}+\frac{2183}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\frac{838 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{691 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}+\frac{1052}{271}+\frac{2183 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(\frac{361}{542}-\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{243 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}-\frac{243 \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)-\frac{29}{18}\right) \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)+\frac{19}{9}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{542}+\frac{437}{1084}+\frac{1653 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{361 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\left(\frac{691}{542}-\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}-\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{2407}{1084}+\frac{159 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\frac{2407 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{691 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}+\frac{719}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(\frac{243 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}+\frac{61 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{361}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\frac{243 \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)-\frac{29}{18}\right) \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)+\frac{19}{9}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{542}-\frac{437}{1084}-\frac{361 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}-\frac{1653 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)^{2}+\left(\frac{243 \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)-\frac{29}{18}\right) \left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)+\frac{19}{9}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}}{542}+\left(-\frac{1511}{1084}+\frac{243 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}-\frac{541 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{1511 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{1653 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}+\frac{310}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{437}{1084}-\frac{361 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}-\frac{1653 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{1511 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{1653 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}+\frac{310}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}+\frac{3489}{1084}+\frac{310 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(\frac{87 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}-\frac{339}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{465}{1084}-\frac{339 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{465}{1084}-\frac{339 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{465 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)}{1084}-\frac{879}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{-n}+\left(\left(\left(\frac{87 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}-2 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}+\frac{3157}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\frac{5863}{1084}-\frac{339 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}}{271}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{874 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(\frac{5863}{1084}-\frac{339 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}}{271}+\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{874 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)-\frac{465 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{719 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{4}}{271}+\frac{719 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{271}-\frac{1438 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{9497}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{691}{542}-\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}-\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{3}-\left(-\frac{691}{542}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}+\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \left(1+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{393}{1084}-\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}+\frac{89 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{3521 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\frac{393 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{691 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}+\frac{691 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}+\frac{3099}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\left(\frac{985}{542}-\frac{437 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}-\frac{747 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{3}-\frac{437 \left(\frac{747 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{874}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}-\frac{985}{874}\right) \left(1+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}}{271}+\left(\frac{4619 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{747 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}+\frac{119 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{271}+\frac{1923}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\frac{985 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}+\frac{985 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{3}}{542}+\frac{4629}{1084}+\frac{1923 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{691}{542}+\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}+\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{2407}{1084}-\frac{159 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{2407 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{691 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}-\frac{719}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)^{2}+\left(\left(-\frac{2407}{1084}-\frac{159 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}+\frac{513 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{2733}{1084}-\frac{159 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}-\frac{363 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{2733 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{2407 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}+\frac{595}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(-\frac{2407 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{691 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{542}-\frac{719}{1084}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{2733 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{1084}-\frac{2407 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}+\frac{595}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{719 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)^{2}}{1084}+\frac{5193}{1084}+\frac{595 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{271}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)^{-n}-\frac{461 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =5\right)^{-n}}{542}\right) \left(\left(\left(\left(\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)-\frac{29}{18}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{18}-\frac{1}{4}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\left(-\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{18}-\frac{1}{4}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{4}+\frac{29}{72}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =3\right)+\left(\left(-\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{18}-\frac{1}{4}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)-\frac{\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{4}+\frac{29}{72}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =4\right)+\left(-\frac{\mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{4}+\frac{29}{72}\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =2\right)+\frac{29 \mathit{RootOf}\left(Z^{5}+Z^{4}-2 Z^{3}+3 Z -1, \mathit{index} =1\right)}{72}-\frac{113}{36}\right) n}{212521} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 101 rules.

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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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \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_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{31}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{16}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 0\\ F_{45}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{47}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{51}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{16}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 0\\ F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{16}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{82}\! \left(x \right)+F_{83}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{16}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{16}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{64}\! \left(x \right)+F_{82}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{16}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{16}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{16}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{83}\! \left(x \right)+F_{91}\! \left(x \right)\\ \end{align*}\)