Av(1234, 1342, 2413, 3142)
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Generating Function
\(\displaystyle -\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}{x^{8}-5 x^{7}+17 x^{6}-36 x^{5}+52 x^{4}-47 x^{3}+26 x^{2}-8 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 203, 621, 1891, 5770, 17653, 54086, 165778, 508108, 1557197, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}-5 x^{7}+17 x^{6}-36 x^{5}+52 x^{4}-47 x^{3}+26 x^{2}-8 x +1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 203\)
\(\displaystyle a \! \left(7\right) = 621\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)+5 a \! \left(n +1\right)-17 a \! \left(n +2\right)+36 a \! \left(n +3\right)-52 a \! \left(n +4\right)+47 a \! \left(n +5\right)-26 a \! \left(n +6\right)+8 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +6}}{32819431}-\frac{7419665 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +6}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +5}}{32819431}+\frac{32425002 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +5}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +4}}{32819431}-\frac{104746494 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +4}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +3}}{32819431}+\frac{198697213 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +3}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +2}}{32819431}-\frac{253675538 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +2}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +1}}{32819431}+\frac{179784201 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n -1}}{32819431}+\frac{12878138 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n -1}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n}}{32819431}-\frac{69021328 \mathit{RootOf} \left(Z^{8}-5 Z^{7}+17 Z^{6}-36 Z^{5}+52 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n}}{32819431}\)
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This specification was found using the strategy pack "Point Placements" and has 67 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_{19}\! \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_{14}\! \left(x \right)+F_{7}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{32}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \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_{35}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \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_{35}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{49}\! \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_{46}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{62}\! \left(x \right)\\ \end{align*}\)