Av(1234, 1342, 2143, 2413, 3124)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{4}}{x^{8}+4 x^{6}-2 x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 149, 412, 1145, 3183, 8841, 24551, 68189, 189418, 526188, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}+4 x^{6}-2 x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 149\)
\(\displaystyle a \! \left(7\right) = 412\)
\(\displaystyle a \! \left(n \right) = -4 a \! \left(n +2\right)+2 a \! \left(n +3\right)+3 a \! \left(n +4\right)-9 a \! \left(n +5\right)+9 a \! \left(n +6\right)-5 a \! \left(n +7\right)+a \! \left(n +8\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +6}}{2942255215}+\frac{313177464 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +6}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +5}}{2942255215}+\frac{109493316 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +5}}{2942255215}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +4}}{588451043}+\frac{242013952 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +4}}{588451043}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +3}}{2942255215}-\frac{291164438 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +3}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +2}}{2942255215}-\frac{1337756324 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +2}}{2942255215}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +1}}{588451043}+\frac{444501677 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +1}}{588451043}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n -1}}{2942255215}+\frac{610144891 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n -1}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n}}{2942255215}-\frac{1379236326 \mathit{RootOf} \left(Z^{8}+4 Z^{6}-2 Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n}}{2942255215}\)
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This specification was found using the strategy pack "Point Placements" and has 70 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_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \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_{60}\! \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_{31}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \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_{7}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{55}\! \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_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)\\ \end{align*}\)