Av(12435, 12453, 14235, 14253, 14325, 14352, 14523, 14532, 21435, 21453, 24135, 24153, 24315, 41235, 41253, 41325, 41352, 41523, 41532, 42135, 42153, 42315, 43125, 43152, 43215)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{3}}{12 x^{8}-10 x^{7}+23 x^{6}-37 x^{5}+6 x^{4}+26 x^{3}-23 x^{2}+8 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 354, 1246, 4266, 14486, 49291, 168461, 577605, 1983135, 6809821, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(12 x^{8}-10 x^{7}+23 x^{6}-37 x^{5}+6 x^{4}+26 x^{3}-23 x^{2}+8 x -1\right) F \! \left(x \right)-\left(x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 354\)
\(\displaystyle a(7) = 1246\)
\(\displaystyle a{\left(n + 8 \right)} = 12 a{\left(n \right)} - 10 a{\left(n + 1 \right)} + 23 a{\left(n + 2 \right)} - 37 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)} + 26 a{\left(n + 5 \right)} - 23 a{\left(n + 6 \right)} + 8 a{\left(n + 7 \right)}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n}}{27313805423}-\frac{35875210301 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n}}{27313805423}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n -1}}{2485556293493}+\frac{623266909448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n -1}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +6}}{2485556293493}+\frac{4434484175064 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n +6}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +5}}{2485556293493}-\frac{897714697812 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n +5}}{2485556293493}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +4}}{191196637961}+\frac{594908699570 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n +4}}{191196637961}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +3}}{2485556293493}-\frac{8886205318448 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +3}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +2}}{2485556293493}-\frac{3844213516999 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n +2}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +1}}{2485556293493}+\frac{7398626152616 \mathit{RootOf} \left(12 Z^{8}-10 Z^{7}+23 Z^{6}-37 Z^{5}+6 Z^{4}+26 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =8\right)^{-n +1}}{2485556293493}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 152 rules.

Finding the specification took 186 seconds.

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Copy 152 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_{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_{29}\! \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_{18}\! \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_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \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_{16}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \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_{40}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \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_{16}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{67}\! \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_{75}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{16}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{69}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{16}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{16}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{16}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{85}\! \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_{10}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{16}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{50}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{16}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{20}\! \left(x \right)+F_{58}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{57}\! \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_{95}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{107}\! \left(x \right)+F_{20}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{129}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{118}\! \left(x \right)+F_{126}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{123}\! \left(x \right)+F_{20}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{135}\! \left(x \right)+F_{139}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{128}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{145}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{146}\! \left(x \right)+F_{20}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{150}\! \left(x \right)+F_{20}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{148}\! \left(x \right)\\ \end{align*}\)