Av(13425, 13452, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15342, 15423, 15432, 31425, 31452, 31542, 34125, 34152, 34215, 41235, 41253, 41325, 41352, 41523, 41532, 42135, 42153, 42315, 43125, 43152, 43215)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x +1\right) \left(x^{2}+x -1\right)^{3}}{2 x^{14}+2 x^{13}-25 x^{12}-51 x^{11}+15 x^{10}+75 x^{9}-5 x^{8}-46 x^{7}-2 x^{6}+15 x^{5}+11 x^{4}-9 x^{3}-x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 90, 316, 1009, 3210, 10303, 33503, 109458, 357559, 1165495, 3794721, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{14}+2 x^{13}-25 x^{12}-51 x^{11}+15 x^{10}+75 x^{9}-5 x^{8}-46 x^{7}-2 x^{6}+15 x^{5}+11 x^{4}-9 x^{3}-x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x +1\right) \left(x^{2}+x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 316\)
\(\displaystyle a(7) = 1009\)
\(\displaystyle a(8) = 3210\)
\(\displaystyle a(9) = 10303\)
\(\displaystyle a(10) = 33503\)
\(\displaystyle a(11) = 109458\)
\(\displaystyle a(12) = 357559\)
\(\displaystyle a(13) = 1165495\)
\(\displaystyle a{\left(n + 14 \right)} = 2 a{\left(n \right)} + 2 a{\left(n + 1 \right)} - 25 a{\left(n + 2 \right)} - 51 a{\left(n + 3 \right)} + 15 a{\left(n + 4 \right)} + 75 a{\left(n + 5 \right)} - 5 a{\left(n + 6 \right)} - 46 a{\left(n + 7 \right)} - 2 a{\left(n + 8 \right)} + 15 a{\left(n + 9 \right)} + 11 a{\left(n + 10 \right)} - 9 a{\left(n + 11 \right)} - a{\left(n + 12 \right)} + 4 a{\left(n + 13 \right)}, \quad n \geq 14\)
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Explicit Closed Form
\(\displaystyle \frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =9\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =10\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n}}{7030513893603156091836020906354}+\frac{748356145971583844697621795341 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n}}{7030513893603156091836020906354}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =9\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =10\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n +12}}{3515256946801578045918010453177}+\frac{174058378219993776343335415707 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n +12}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =10\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +11}}{3515256946801578045918010453177}+\frac{286150871202083865613811767011 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =9\right)^{-n +11}}{3515256946801578045918010453177}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =9\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =10\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n +10}}{7030513893603156091836020906354}-\frac{3945866342038208218389480706357 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n +10}}{7030513893603156091836020906354}-\frac{11459356759627671222711800727367 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +9}}{7030513893603156091836020906354}-\frac{11459356759627671222711800727367 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +9}}{7030513893603156091836020906354}-\frac{11459356759627671222711800727367 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +9}}{7030513893603156091836020906354}-\frac{11459356759627671222711800727367 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, 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\mathit{index} =2\right)^{-n +6}}{7030513893603156091836020906354}+\frac{7588326038084173094352345482283 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +6}}{7030513893603156091836020906354}+\frac{7588326038084173094352345482283 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +6}}{7030513893603156091836020906354}+\frac{7588326038084173094352345482283 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +6}}{7030513893603156091836020906354}+\frac{7588326038084173094352345482283 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, 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\mathit{index} =10\right)^{-n +2}}{3515256946801578045918010453177}+\frac{531263699058756503321492617401 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n +2}}{3515256946801578045918010453177}+\frac{531263699058756503321492617401 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n +2}}{3515256946801578045918010453177}+\frac{531263699058756503321492617401 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n +2}}{3515256946801578045918010453177}+\frac{531263699058756503321492617401 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n +2}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =9\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =10\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n +1}}{3515256946801578045918010453177}+\frac{61209178238897993542899723864 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n +1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =9\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =10\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =11\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =12\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =13\right)^{-n -1}}{3515256946801578045918010453177}+\frac{143039812504856192928145019655 \mathit{RootOf} \left(2 Z^{14}+2 Z^{13}-25 Z^{12}-51 Z^{11}+15 Z^{10}+75 Z^{9}-5 Z^{8}-46 Z^{7}-2 Z^{6}+15 Z^{5}+11 Z^{4}-9 Z^{3}-Z^{2}+4 Z -1, \mathit{index} =14\right)^{-n -1}}{3515256946801578045918010453177}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 218 rules.

Finding the specification took 181 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_{14}\! \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_{34}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{18}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{61}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{14}\! \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_{60}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{14}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{14}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{73}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{14}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= x^{2}\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{145}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{126}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{124}\! \left(x \right)+F_{18}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{123}\! \left(x \right) &= 0\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{134}\! \left(x \right)+F_{141}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{137}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{130}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{14}\! \left(x \right) F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{14}\! \left(x \right) F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{158}\! \left(x \right)+F_{18}\! \left(x \right)+F_{190}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{14}\! \left(x \right) F_{147}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{149}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{150}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{14}\! \left(x \right) F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{154}\! \left(x \right)\\ F_{153}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{154}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{155}\! \left(x \right)+F_{156}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{155}\! \left(x \right) &= 0\\ F_{156}\! \left(x \right) &= F_{14}\! \left(x \right) F_{157}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{14}\! \left(x \right) F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{189}\! \left(x \right)\\ F_{162}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{163}\! \left(x \right)+F_{169}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{14}\! \left(x \right) F_{164}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{166}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{150}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{168}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{155}\! \left(x \right)+F_{156}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{14}\! \left(x \right) F_{170}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{172}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{163}\! \left(x \right)+F_{169}\! \left(x \right)\\ F_{173}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{174}\! \left(x \right)+F_{178}\! \left(x \right)+F_{185}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{14}\! \left(x \right) F_{175}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{177}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{14}\! \left(x \right) F_{179}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{181}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{174}\! \left(x \right)+F_{178}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{14}\! \left(x \right) F_{184}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{181}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{14}\! \left(x \right) F_{186}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{188}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{181}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{181}\! \left(x \right)\\ F_{189}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{174}\! \left(x \right)+F_{178}\! \left(x \right)+F_{185}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{14}\! \left(x \right) F_{191}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)+F_{193}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{206}\! \left(x \right)\\ F_{194}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{195}\! \left(x \right)+F_{200}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{14}\! \left(x \right) F_{196}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{198}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{118}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{14}\! \left(x \right) F_{201}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{199}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)+F_{205}\! \left(x \right)\\ F_{204}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{130}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{139}\! \left(x \right)\\ F_{206}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{207}\! \left(x \right)+F_{212}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{14}\! \left(x \right) F_{208}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{210}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{149}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{211}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{14}\! \left(x \right) F_{213}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{214}\! \left(x \right)+F_{215}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{216}\! \left(x \right)+F_{217}\! \left(x \right)\\ F_{216}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{174}\! \left(x \right)+F_{178}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{183}\! \left(x \right)\\ \end{align*}\)