Av(1342, 2143, 2413, 3124)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}{4 x^{8}-4 x^{7}-12 x^{6}+47 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 206, 643, 1992, 6153, 18991, 58614, 180929, 558536, 1724282, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{8}-4 x^{7}-12 x^{6}+47 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 206\)
\(\displaystyle a \! \left(7\right) = 643\)
\(\displaystyle a \! \left(n +8\right) = 4 a \! \left(n \right)-4 a \! \left(n +1\right)-12 a \! \left(n +2\right)+47 a \! \left(n +3\right)-74 a \! \left(n +4\right)+65 a \! \left(n +5\right)-33 a \! \left(n +6\right)+9 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +6}}{3145}+\frac{3752 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +6}}{3145}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +5}}{22015}-\frac{22628 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +5}}{22015}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +4}}{629}-\frac{2524 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +4}}{629}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +3}}{22015}+\frac{294318 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +3}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +2}}{22015}-\frac{425521 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +2}}{22015}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +1}}{4403}+\frac{63942 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +1}}{4403}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n -1}}{22015}+\frac{19622 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n -1}}{22015}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n}}{4403}-\frac{24631 \mathit{RootOf} \left(4 Z^{8}-4 Z^{7}-12 Z^{6}+47 Z^{5}-74 Z^{4}+65 Z^{3}-33 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n}}{4403}\)
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This specification was found using the strategy pack "Point Placements" and has 56 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_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \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_{48}\! \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_{28}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \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_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \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_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \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_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \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_{47}\! \left(x \right)\\ F_{43}\! \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_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{35}\! \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_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)\\ \end{align*}\)