Av(1234, 1342, 1423, 2143, 3214)
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Generating Function
\(\displaystyle -\frac{x^{3}+x^{2}+x -1}{2 x^{8}+4 x^{7}+4 x^{6}-x^{5}-3 x^{4}-2 x^{3}-x^{2}-2 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 52, 138, 378, 1047, 2889, 7943, 21845, 60132, 165541, 455646, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{8}+4 x^{7}+4 x^{6}-x^{5}-3 x^{4}-2 x^{3}-x^{2}-2 x +1\right) F \! \left(x \right)+x^{3}+x^{2}+x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 138\)
\(\displaystyle a \! \left(7\right) = 378\)
\(\displaystyle a \! \left(n +8\right) = -2 a \! \left(n \right)-4 a \! \left(n +1\right)-4 a \! \left(n +2\right)+a \! \left(n +3\right)+3 a \! \left(n +4\right)+2 a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n +6}}{54555387391}-\frac{1344869652 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n +6}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n +5}}{54555387391}-\frac{3051180216 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n +5}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n +4}}{54555387391}-\frac{5070936826 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n +4}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n +3}}{54555387391}-\frac{5217027252 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n +3}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n +2}}{54555387391}-\frac{899190950 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n +2}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n +1}}{54555387391}+\frac{4059297833 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n +1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n -1}}{54555387391}+\frac{2381508199 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n -1}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =1\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =2\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =3\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =4\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =5\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =6\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =7\right)^{-n}}{54555387391}+\frac{9805626918 \mathit{RootOf} \left(2 Z^{8}+4 Z^{7}+4 Z^{6}-Z^{5}-3 Z^{4}-2 Z^{3}-Z^{2}-2 Z +1, \mathit{index} =8\right)^{-n}}{54555387391}\)
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This specification was found using the strategy pack "Point Placements" and has 71 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{59}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= x^{2}\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)