Av(1243, 1342, 2134, 2413, 3124)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{2}}{x^{8}+2 x^{7}+x^{6}-x^{5}-x^{4}+6 x^{3}-8 x^{2}+5 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 167, 488, 1426, 4163, 12144, 35418, 103302, 301320, 878954, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}+2 x^{7}+x^{6}-x^{5}-x^{4}+6 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 167\)
\(\displaystyle a \! \left(7\right) = 488\)
\(\displaystyle a \! \left(n +8\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)-a \! \left(n +3\right)-a \! \left(n +4\right)+6 a \! \left(n +5\right)-8 a \! \left(n +6\right)+5 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +6}}{6565172363}+\frac{352012550 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +6}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +5}}{6565172363}+\frac{775052961 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +5}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +4}}{6565172363}+\frac{380434700 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +4}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +3}}{6565172363}-\frac{652934211 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +3}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +2}}{6565172363}-\frac{865759159 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +2}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +1}}{6565172363}+\frac{1986763722 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n -1}}{6565172363}+\frac{930863637 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n -1}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n}}{6565172363}-\frac{1491669994 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+Z^{6}-Z^{5}-Z^{4}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n}}{6565172363}\)
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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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \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_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \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_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{53}\! \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_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{68}\! \left(x \right)\\ \end{align*}\)