Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 24513, 31425, 31524, 32415, 32514)
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Generating Function
\(\displaystyle -\frac{\left(3 x -1\right) \left(x -1\right)^{2}}{x^{5}-2 x^{4}-8 x^{3}+11 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 357, 1299, 4669, 16747, 60134, 216180, 777621, 2797643, 10064988, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-2 x^{4}-8 x^{3}+11 x^{2}-6 x +1\right) F \! \left(x \right)+\left(3 x -1\right) \left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = - a{\left(n \right)} + 2 a{\left(n + 1 \right)} + 8 a{\left(n + 2 \right)} - 11 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{287002562368 \left(\left(\left(\left(-1+\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{8918561}{2690111}-\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{4438308}{2690111}+\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-1+\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{22277570}{2690111}-\frac{7303742 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{76105347}{2690111}+\frac{13976097 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{36732222}{2690111}-\frac{6317856 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{8918561}{2690111}-\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{76105347}{2690111}+\frac{13976097 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{136280074}{2690111}+\frac{1170566 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{64554483}{2690111}+\frac{764634 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{4438308}{2690111}+\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{36732222}{2690111}-\frac{6317856 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{64554483}{2690111}+\frac{764634 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{8843271 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}+\frac{62333412}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}+\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{939774}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{11288291 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}+\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}+\frac{6220597}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{3104317}{5380222}-\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}+\frac{6220597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{11288291 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}+\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}+\frac{6220597}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{11288291 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}-\frac{36415886}{2690111}+\frac{151239339 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{10272351}{5380222}-\frac{36415886 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+\frac{6220597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{3104317}{5380222}-\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}+\frac{6220597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{10272351}{5380222}-\frac{36415886 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+\frac{6220597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{10272351 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}-\frac{3104317 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}+\frac{40583153}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(-\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{7978787}{2690111}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{5244861 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{13359009}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{65349183}{2690111}-\frac{13359009 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{4838778 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{5563314}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{34952922}{2690111}-\frac{5563314 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{8854496 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+\frac{63236506}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{439482607}{2690111}+\frac{63236506 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{44349735}{2690111}-\frac{593790 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{285178260}{2690111}-\frac{44349735 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-1+\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{8918561}{2690111}-\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{4438308}{2690111}+\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{8918561}{2690111}-\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{11288291 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}-\frac{113880431}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{58815529}{5380222}-\frac{6220597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{4438308}{2690111}+\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{58815529}{5380222}-\frac{6220597 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{21750259}{5380222}+\frac{3104317 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(-\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{7978787}{2690111}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{5244861 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{13359009}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{65349183}{2690111}-\frac{13359009 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{16663903 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}+\frac{38330263}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{212215351}{5380222}+\frac{38330263 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{14648915}{5380222}+\frac{6415115 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{89672165}{5380222}-\frac{14648915 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(-\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{7978787}{2690111}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2058881 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{8918561}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{53829917}{2690111}-\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{4438308}{2690111}-\frac{939774 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{27855606}{2690111}+\frac{4438308 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{44426509 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{30185011}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{43416922}{2690111}-\frac{30185011 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{43416922}{2690111}-\frac{30185011 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{258129439}{2690111}+\frac{43416922 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{26341459 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}+\frac{49456891}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{142309507}{5380222}+\frac{49456891 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{142309507}{5380222}+\frac{49456891 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{879937909}{5380222}-\frac{142309507 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{3185980 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+\frac{4440448}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{11519266}{2690111}+\frac{4440448 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{11519266}{2690111}+\frac{4440448 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{70927612}{2690111}-\frac{11519266 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(\frac{1277375 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-1\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{7978787}{2690111}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{7978787}{2690111}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{49391609}{2690111}+\frac{7978787 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{7978787}{2690111}-\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{53829917}{2690111}+\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{27855606}{2690111}-\frac{4438308 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}-\left(\left(-\frac{7978787}{2690111}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{53829917}{2690111}-\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{27855606}{2690111}+\frac{4438308 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-2\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{53829917}{2690111}+\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{135515440}{2690111}-\frac{22275430 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{577418607}{5380222}-\frac{92811267 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{325310965}{5380222}+\frac{44382019 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{27855606}{2690111}-\frac{4438308 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{55711212}{2690111}+\frac{8876616 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{325310965}{5380222}+\frac{44382019 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{35275703 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)}{5380222}-\frac{210134963}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}-\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}+\frac{4438308}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{113880431 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}-\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}-\frac{58815529}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{21750259}{5380222}+\frac{4438308 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}-\frac{58815529 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{113880431 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}-\frac{8918561 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}-\frac{58815529}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{113880431 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}+\frac{91881668}{2690111}-\frac{438203187 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{8224815}{5380222}+\frac{91881668 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{58815529 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{21750259}{5380222}+\frac{4438308 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2690111}-\frac{58815529 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{8224815}{5380222}+\frac{91881668 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}-\frac{58815529 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{8224815 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}+\frac{21750259 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{5380222}-\frac{342481813}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\frac{7978787}{2690111}+\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{49391609}{2690111}-\frac{7978787 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-2 \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{15957574}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{98783218}{2690111}+\frac{15957574 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-8 \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{63830296}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{395132872}{2690111}+\frac{63830296 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2690111}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{32132477 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}-\frac{172437641}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{1116628907}{5380222}-\frac{172437641 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{120378731}{5380222}-\frac{26630909 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{771578147}{5380222}+\frac{120378731 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{27049965 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}+\frac{3095673}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{30013509}{5380222}+\frac{3095673 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{30013509}{5380222}+\frac{3095673 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{194836413}{5380222}+\frac{30013509 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{5380222}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}-\frac{173149497 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{5380222}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)-\frac{14313}{26672}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{14313 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{26672}+\frac{3447}{26672}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{14313 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{26672}+\frac{3447}{26672}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{3447 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{26672}+\frac{24463}{53344}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{14313 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{26672}+\frac{3447}{26672}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{3447 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{26672}+\frac{24463}{53344}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{3447 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{26672}+\frac{24463}{53344}\right) \mathit{RootOf} \left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{24463 \mathit{RootOf}\left(Z^{5}-2 Z^{4}-8 Z^{3}+11 Z^{2}-6 Z +1, \mathit{index} =3\right)}{53344}+\frac{99303}{53344}\right)}{144834532905087}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 92 rules.

Finding the specification took 103 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_{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_{31}\! \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_{22}\! \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_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \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_{17}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{40}\! \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_{49}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{16}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{63}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{16}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\ 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_{2}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{62}\! \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_{73}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \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_{77}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{63}\! \left(x \right)+F_{79}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{16}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{81}\! \left(x \right) &= 0\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= 0\\ F_{85}\! \left(x \right) &= F_{16}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{75}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{16}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{31}\! \left(x \right)\\ \end{align*}\)