Av(1234, 1243, 1342, 2413, 4213)
Generating Function
\(\displaystyle \frac{3 x^{5}-3 x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{3}+2 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 139, 352, 868, 2098, 4995, 11753, 27395, 63368, 145656, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x -1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)-3 x^{5}+3 x^{4}-x^{3}+5 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+1, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{97 \left(\frac{5664 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\frac{5}{3}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}-\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} 576^{n} \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}-18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{97}+\frac{9440 \,576^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}-\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(-2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}+18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{97}-\frac{39648 \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} 576^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}-\frac{9 \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}+\frac{3 \,\mathrm{I} \sqrt{3}}{8}-\frac{3}{8}\right)^{-n} \left(-\frac{3 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{4}-\frac{3 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{4}+\frac{\left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) 18^{\frac{1}{3}}+27 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{32}\right)^{n}}{97}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(-\frac{118 \left(\frac{149 \sqrt{59}}{177}+\mathrm{I}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{97}-\frac{2065 \left(-\frac{97 \sqrt{59}}{2065}+\mathrm{I}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{582}+\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(3^{2 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I} \sqrt{59}-\frac{2065}{291}\right) 2^{6 n +\frac{1}{3}}+\frac{298 \,2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{2 n +\frac{1}{3}} \left(\mathrm{I} \sqrt{59}+\frac{59}{149}\right)}{97}\right)\right)-\left(-\frac{9440 \,576^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}-\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}+3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(-256 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-256 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-32 \,\mathrm{I} \sqrt{59}-96\right) 18^{\frac{1}{3}}+288 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+32 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{97}-\frac{596 \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\sqrt{59}\, 1024^{n} 3^{3 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{97 \sqrt{59}\, 1024^{n} 3^{3 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{596}-\frac{177 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(3^{3 n +\frac{1}{3}} 2^{10 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{35 \,3^{3 n +\frac{2}{3}} 2^{10 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{6}\right)}{149}\right) \left(8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}+3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{291}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(-\frac{118 \left(\mathrm{I}-\frac{149 \sqrt{59}}{177}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{97}-\frac{2065 \left(\mathrm{I}+\frac{97 \sqrt{59}}{2065}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{582}+\left(3^{2 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{2065}{291}\right) 2^{6 n +\frac{1}{3}}+\frac{298 \,2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\mathrm{I} \sqrt{59}-\frac{59}{149}\right) 3^{2 n +\frac{1}{3}}}{97}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}\right)\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}}{11328} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 82 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_{21}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \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_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
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_{51}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{35}\! \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_{11}\! \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_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \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_{4}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{56}\! \left(x \right)\\
\end{align*}\)