Av(1243, 1432, 2134, 3214)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{4 x^{5}-6 x^{3}+8 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 60, 172, 492, 1420, 4116, 11932, 34564, 100092, 289860, 839484, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{5}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = \frac{3 a \! \left(n +2\right)}{2}-2 a \! \left(n +3\right)+\frac{5 a \! \left(n +4\right)}{4}-\frac{a \! \left(n +5\right)}{4}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = \frac{3 a \! \left(n +2\right)}{2}-2 a \! \left(n +3\right)+\frac{5 a \! \left(n +4\right)}{4}-\frac{a \! \left(n +5\right)}{4}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{9687744 \left(\left(\left(\left(\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)-\frac{71}{121}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{71 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{121}+\frac{607}{1452}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{71 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{121}+\frac{607}{1452}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{607 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1452}-\frac{133}{484}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\left(-\frac{71 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{121}+\frac{607}{1452}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{607 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1452}-\frac{133}{484}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{607 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1452}-\frac{133}{484}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{133 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{484}-\frac{581}{2904}\right) \left(\left(\left(\left(\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{139}-1\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}+\frac{1059}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{925}{6672}-\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{139}-1\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-\frac{11015 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}+\frac{2419}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{379 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1112}-\frac{3923}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1147}{2224}+\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}+\frac{1059}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{379 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1112}-\frac{3923}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{14667}{4448}+\frac{76687 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{16961 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4448}+\frac{34769}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{925}{6672}-\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1147}{2224}+\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{16961 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4448}+\frac{34769}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{9121}{4448}+\frac{34769 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}-\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{139}+\frac{301}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{6672}-\frac{119 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{278}-\frac{21}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{63}{1112}-\frac{21 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}+\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{6672}-\frac{119 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{278}-\frac{21}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{119 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{278}-\frac{19759}{4448}+\frac{93851 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{21 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}-\frac{19759 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4448}+\frac{41077}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{63}{1112}-\frac{21 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}+\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{21 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}-\frac{19759 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4448}+\frac{41077}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{9121}{4448}-\frac{63 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1112}+\frac{41077 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13344}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{85}{139}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(-\frac{85}{139}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{202}{139}-\frac{85 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{3}{2}+\frac{258 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{255}{278}-\frac{3 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1723}{556}+\frac{3863 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{834}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1057}{1668}-\frac{1723 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{556}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{4693}{1668}-\frac{2233 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{556}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{4693 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1668}-\frac{355}{278}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{139}-1\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}+\frac{1059}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{925}{6672}-\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}+\frac{1059}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{2177}{1668}+\frac{119 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{278}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1273}{2224}+\frac{21 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{925}{6672}-\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1273}{2224}+\frac{21 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{63 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{1112}-\frac{1577}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{85}{139}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+\left(\left(-\frac{85}{139}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{202}{139}-\frac{85 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{431}{3336}+\frac{97 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1112}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{63}{1112}-\frac{431 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{63}{1112}-\frac{431 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{63 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1112}-\frac{1577}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{85}{139}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1059}{2224}+\frac{4343 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{8771}{6672}-\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{925}{6672}+\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}-\frac{1147}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(-\frac{3345}{556}+\frac{3589 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{417}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{6733}{1668}-\frac{3345 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{556}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{6733}{1668}-\frac{3345 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{556}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{6733 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1668}-\frac{1163}{278}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(\frac{4573}{3336}-\frac{1967 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1112}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{957}{1112}+\frac{4573 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{957}{1112}+\frac{4573 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{957 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1112}+\frac{5695}{3336}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(\frac{301}{2224}-\frac{2329 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{925}{6672}+\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{925}{6672}+\frac{301 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}-\frac{1147}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{172 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{85}{139}-\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\frac{85}{139}-\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{85 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}-\frac{202}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(-\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-\frac{925}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}+\frac{2177 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1668}-\frac{1273}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1577}{3336}-\frac{1273 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}-\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}+\frac{2177 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{1668}-\frac{1273}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{2177 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{1668}+\frac{39655}{13344}-\frac{22519 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{1273 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}+\frac{39655 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13344}-\frac{7709}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{1577}{3336}-\frac{1273 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}-\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1273 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{2224}+\frac{39655 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{13344}-\frac{7709}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{18221}{13344}+\frac{1577 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}}{3336}-\frac{7709 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}+\left(\left(\left(\frac{85}{139}-\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}-\frac{8771}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{1147}{2224}+\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-\left(\left(\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{85}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}+\frac{8771}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)-\frac{1147}{2224}-\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{1059 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{2224}-\frac{8771}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{1147}{2224}+\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\frac{58667}{13344}-\frac{17427 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{33347 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13344}-\frac{12019}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(\frac{1147}{2224}+\frac{925 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(\frac{33347 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{13344}-\frac{12019}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\frac{9345}{4448}-\frac{7709 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)}{4448}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}+\left(\left(\left(-\frac{85}{139}+\mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{202}{139}-\frac{85 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{139}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+\left(\left(\frac{255}{278}-\frac{3 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{303}{139}+\frac{255 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{278}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(\left(\frac{16177}{6672}-\frac{7125 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)+\frac{1031}{2224}+\frac{16177 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-\frac{4289}{2224}+\frac{18061 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{4289 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}+\frac{6205}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}+\left(\left(\left(\frac{24337}{6672}-\frac{11573 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{5433}{2224}+\frac{24337 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-\frac{5433}{2224}+\frac{24337 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)-\frac{5433 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)}{2224}+\frac{17305}{6672}\right) \mathit{RootOf} \left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}+\frac{2591 \mathit{RootOf}\left(4 Z^{5}-6 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{6672}\right)}{6713281}\)
This specification was found using the strategy pack "Point Placements" and has 83 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_{20}\! \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_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{66}\! \left(x \right)\\
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_{38}\! \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_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{64}\! \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_{42}\! \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_{51}\! \left(x \right)+F_{62}\! \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_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \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_{58}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{70}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= x^{2}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
\end{align*}\)