Av(12435, 12453, 14235, 14253, 14325, 14352, 14523, 14532, 21435, 21453, 24135, 24153, 41235, 41253, 41325, 41352, 41523, 41532, 42135, 42153)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x +1\right) \left(5 x^{2}-5 x +1\right)}{2 x^{5}-12 x^{4}+3 x^{3}+8 x^{2}-6 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 412, 1668, 6688, 26700, 106436, 424144, 1690196, 6735740, 26844272, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-12 x^{4}+3 x^{3}+8 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(x +1\right) \left(5 x^{2}-5 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = - 2 a{\left(n \right)} + 12 a{\left(n + 1 \right)} - 3 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 5 \right)} = - 2 a{\left(n \right)} + 12 a{\left(n + 1 \right)} - 3 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{74087345431 \left(\left(\left(\left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)-\frac{37915}{66026}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{37915 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{66026}+\frac{15593}{66026}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{37915 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{66026}+\frac{15593}{66026}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{15593 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{66026}-\frac{45939}{132052}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{37915 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{66026}+\frac{15593}{66026}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{15593 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{66026}-\frac{45939}{132052}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{15593 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{66026}-\frac{45939}{132052}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{45939 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{132052}+\frac{299549}{132052}\right) \left(\left(\left(\left(-\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-1\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{53987}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{62178}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-1\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{2057232 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{10631288}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{8186395 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{3701681}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{4932630 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{4246156}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{53987}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{8186395 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{3701681}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{4573281 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{12858075}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{6410039 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{10894990}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{62178}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{4932630 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{4246156}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{6410039 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{10894990}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{1681010 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{261595}{204017}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{998276}{2244187}+\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2244187}+\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{263383}{4488374}+\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2244187}-\frac{1313775 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{263383 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2244187}+\frac{1101223}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{263383}{4488374}+\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2244187}-\frac{1313775 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2140309 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{408034}-\frac{8194965}{2244187}-\frac{1313775 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{263383 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}-\frac{8194965 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{63043}{408034}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{263383 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{2244187}+\frac{1101223}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{263383 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}-\frac{8194965 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{63043}{408034}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{63043 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{408034}+\frac{1101223 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}-\frac{4777990}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1+\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{11549067}{2244187}-\frac{3408215 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{11549067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{7577285}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{16260099}{4488374}-\frac{24104043 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{16260099 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{41280375}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{237961}{408034}+\frac{12051423 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{237961 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{408034}+\frac{11948077}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{1565297 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{95471}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{95471 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{266455}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(-\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-1\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{53987}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{62178}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}-\frac{53987}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{544311}{408034}+\frac{1313775 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{263383 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}+\frac{3090777}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{2244187}+\frac{62178}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{263383 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}+\frac{3090777}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{1101223 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}-\frac{2551845}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1+\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{11549067}{2244187}-\frac{3408215 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{11549067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{7577285}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{13390783}{4488374}-\frac{15059015 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{13390783 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{35445451}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{9601877 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}+\frac{11583089}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{11583089 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}+\frac{22600155}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1+\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{53987}{132011}+\frac{1350983 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{132011}+\frac{2862019}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{998276 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{62178}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{62178 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{132011}-\frac{2096000}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{10542045}{2244187}-\frac{1108837 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{10542045 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{7568749}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\frac{10542045 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{7568749}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{7568749 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{15942635}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{1434658}{2244187}+\frac{4522514 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{1434658 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{2917462}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{1434658 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{2917462}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{2917462 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{5421510}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{12466846}{2244187}+\frac{4759198 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{12466846 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{10439304}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{12466846 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{10439304}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{10439304 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{21418270}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1-\frac{942067 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)-\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)-\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{1916055 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{3919045}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}-\frac{2862019}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{62178 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}+\frac{2096000}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}-\left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-6\right) \left(\left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}+\frac{2862019}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{62178 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}-\frac{2096000}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}-\frac{2862019}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{35100 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{12001}+\frac{19268114}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{206961 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}-\frac{15151401}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{7977895 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}-\frac{6436785}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(\frac{62178 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}+\frac{2096000}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{373068 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{132011}-\frac{12576000}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{7977895 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}-\frac{6436785}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\frac{2614283 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)}{4488374}+\frac{1325147}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{62178}{132011}+\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}+\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{132011}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{3090777}{4488374}+\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{132011}+\frac{544311 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{408034}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{3090777 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{62178 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{132011}+\frac{2551845}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{3090777}{4488374}+\frac{53987 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{132011}+\frac{544311 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{408034}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{42313041 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}+\frac{6559306}{2244187}+\frac{544311 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{408034}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{3090777 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}+\frac{6559306 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{12696787}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{3090777 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{62178 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{132011}+\frac{2551845}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{3090777 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}+\frac{6559306 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{12696787}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)-\frac{12696787 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}+\frac{2551845 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)^{2}}{4488374}-\frac{2903946}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)+\frac{1916055}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{1916055 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{3919045}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{11496330}{2244187}-6 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{11496330 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}-\frac{23514270}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{5748165}{4488374}+\frac{3 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{5748165 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}+\frac{11757135}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{844848}{204017}+\frac{692304 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{204017}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{844848 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{204017}+\frac{16184046}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)+\left(-\frac{1013543 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{408034}-\frac{12724785}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)-\frac{12724785 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{4488374}-\frac{22783285}{4488374}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{1629108}{2244187}-\frac{123764 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{204017}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{1629108 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{507866}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =2\right)+\left(\frac{1629108 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{507866}{2244187}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =4\right)+\frac{507866 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =3\right)}{2244187}+\frac{120320}{204017}\right) \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}-\frac{4476180 \mathit{RootOf}\left(2 Z^{5}-12 Z^{4}+3 Z^{3}+8 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{2244187}\right)}{834841141350}\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 67 rules.
Finding the specification took 90 seconds.
Copy 67 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{15}\! \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_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{15}\! \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_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{23}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{15}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{35}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{15}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{15}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{15}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{55}\! \left(x \right)\\
\end{align*}\)