Av(13425, 13452, 13542, 14325, 14352, 14532, 15342, 15432, 31425, 31452, 31542, 34125, 34152, 41325, 41352, 41532, 43125, 43152)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x +1\right) \left(3 x -1\right)^{2} \left(2 x -1\right)^{3}}{15 x^{10}+65 x^{9}-103 x^{8}+209 x^{7}-248 x^{6}+20 x^{5}+188 x^{4}-169 x^{3}+67 x^{2}-13 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 102, 428, 1729, 6797, 26302, 101048, 387224, 1483196, 5682132, 21772765, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(15 x^{10}+65 x^{9}-103 x^{8}+209 x^{7}-248 x^{6}+20 x^{5}+188 x^{4}-169 x^{3}+67 x^{2}-13 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x +1\right) \left(3 x -1\right)^{2} \left(2 x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a(6) = 428\)
\(\displaystyle a(7) = 1729\)
\(\displaystyle a(8) = 6797\)
\(\displaystyle a(9) = 26302\)
\(\displaystyle a{\left(n + 10 \right)} = - 15 a{\left(n \right)} - 65 a{\left(n + 1 \right)} + 103 a{\left(n + 2 \right)} - 209 a{\left(n + 3 \right)} + 248 a{\left(n + 4 \right)} - 20 a{\left(n + 5 \right)} - 188 a{\left(n + 6 \right)} + 169 a{\left(n + 7 \right)} - 67 a{\left(n + 8 \right)} + 13 a{\left(n + 9 \right)}, \quad n \geq 10\)
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Explicit Closed Form
\(\displaystyle \frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +8}}{15102160831733058536}+\frac{142450988394799864875 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +8}}{15102160831733058536}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +7}}{3775540207933264634}+\frac{175019205756493093765 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +7}}{3775540207933264634}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +6}}{15102160831733058536}-\frac{577448571090281512615 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +6}}{15102160831733058536}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +5}}{7551080415866529268}+\frac{808608530142923765009 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +5}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +4}}{7551080415866529268}-\frac{699793269920249004527 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +4}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +3}}{7551080415866529268}-\frac{344699012953015018529 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +3}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +2}}{7551080415866529268}+\frac{712958788141691015729 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +2}}{7551080415866529268}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n +1}}{15102160831733058536}-\frac{734844524918382332731 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n +1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n -1}}{15102160831733058536}-\frac{12327088719937136801 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n -1}}{15102160831733058536}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =1\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =2\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =3\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =4\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =5\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =6\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =7\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =8\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =9\right)^{-n}}{1887770103966632317}+\frac{20131651592418601750 \mathit{RootOf} \left(15 Z^{10}+65 Z^{9}-103 Z^{8}+209 Z^{7}-248 Z^{6}+20 Z^{5}+188 Z^{4}-169 Z^{3}+67 Z^{2}-13 Z +1, \mathit{index} =10\right)^{-n}}{1887770103966632317}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 122 rules.

Finding the specification took 594 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_{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_{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_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \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_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{12}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \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_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{37}\! \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_{39}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= x^{2}\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{50}\! \left(x \right) &= 0\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{52}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{54}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{59}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \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_{80}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{52}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{16}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{16}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{16}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{16}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{67}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{16}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{53}\! \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_{11}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{16}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{16}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{71}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{16}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{97}\! \left(x \right) &= 0\\ F_{98}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{107}\! \left(x \right)+F_{12}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{107}\! \left(x \right) &= 0\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{119}\! \left(x \right)+F_{12}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{117}\! \left(x \right)+F_{12}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{117}\! \left(x \right) &= 0\\ F_{118}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{121}\! \left(x \right) &= 0\\ \end{align*}\)