Av(1234, 1342, 1423, 2413, 3241)
Generating Function
\(\displaystyle -\frac{5 x^{6}-7 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 156, 417, 1078, 2712, 6675, 16142, 38484, 90695, 211730, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+5 x^{6}-7 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 156\)
\(\displaystyle a \! \left(n +4\right) = n^{2}-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-2 a \! \left(n \right)+a \! \left(n +1\right)-n +4, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 156\)
\(\displaystyle a \! \left(n +4\right) = n^{2}-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-2 a \! \left(n \right)+a \! \left(n +1\right)-n +4, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{469 \left(\frac{2 \left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{1593 \,3^{\frac{1}{3}}}{469}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{657 \,2^{\frac{1}{3}} \left(\sqrt{59}\, 3^{\frac{1}{6}}-\frac{5723 \,3^{\frac{2}{3}}}{657}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{938}\right) \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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \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} \left(-2304 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-2304 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-288 \,\mathrm{I} \sqrt{59}-864\right) 18^{\frac{1}{3}}+2592 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+288 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-384 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-384 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(96 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-288 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(576 \,\mathrm{I}-192 \sqrt{3}\right) \sqrt{59}-1728 \,\mathrm{I} \sqrt{3}+1728\right)^{-n}}{3}+\frac{17936 \left(\frac{\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}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(-\frac{512 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{512 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(-64 \,\mathrm{I} \sqrt{59}-192\right) 18^{\frac{1}{3}}+576 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+64 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n} \left(\frac{\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}}-\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(\left(\mathrm{I} \sqrt{59}-3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}\right)^{n}}{469}+\frac{5664 \left(\frac{\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}}-\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(\left(n^{2}+\frac{13}{2}\right) \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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\frac{19 \,576^{n}}{6}\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} \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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{469}+\frac{17936 \,576^{n} \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} \left(\frac{\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}}-\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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{469}-\frac{11328 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}+\frac{9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{32}\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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \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}}{67}+\left(\frac{531 \left(3^{\frac{5}{6}} \left(\mathrm{I}-\frac{469 \sqrt{59}}{1593}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{97 \,2^{\frac{1}{3}} \left(\mathrm{I}+\frac{219 \sqrt{59}}{5723}\right) 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{18}\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} \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 \sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{469}+\left(\left(-\frac{219 \,3^{-n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{5723}{657}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 2^{-5 n +\frac{1}{3}}}{938}-\left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 2^{-5 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}\, 3^{-n +\frac{1}{3}}-\frac{177 \,3^{\frac{4}{3}-n}}{469}\right)\right) \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(\left(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{59}-\frac{531 \,\mathrm{I} \,3^{\frac{5}{6}}}{469}+\frac{531 \,3^{\frac{1}{3}}}{469}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{219 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{5723 \,\mathrm{I} \,3^{\frac{1}{6}}}{219}-\frac{5723 \,3^{\frac{2}{3}}}{657}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{938}\right)\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{11328}\)
This specification was found using the strategy pack "Point Placements" and has 313 rules.
Found on January 18, 2022.Finding the specification took 15 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_{17}\! \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_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{247}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \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_{11}\! \left(x \right)+F_{27}\! \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_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{52}\! \left(x \right)+F_{74}\! \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_{59}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{60}\! \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_{55}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \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_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{74}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{108}\! \left(x \right)+F_{216}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{111}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{112}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{189}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{134}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{141}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{185}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{156}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{176}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{163}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{169}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{169}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{188}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{186}\! \left(x \right)\\
F_{189}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{190}\! \left(x \right)+F_{212}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)+F_{197}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{193}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{195}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{196}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{203}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{200}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{202}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{198}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{205}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{205}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{207}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{193}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{209}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{209}\! \left(x \right) &= F_{210}\! \left(x \right)+F_{211}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{193}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{208}\! \left(x \right)\\
F_{212}\! \left(x \right) &= F_{213}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{213}\! \left(x \right) &= F_{214}\! \left(x \right)+F_{215}\! \left(x \right)\\
F_{214}\! \left(x \right) &= F_{159}\! \left(x \right)\\
F_{215}\! \left(x \right) &= F_{186}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{217}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{218}\! \left(x \right)+F_{245}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{219}\! \left(x \right) &= F_{220}\! \left(x \right)\\
F_{220}\! \left(x \right) &= F_{221}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{221}\! \left(x \right) &= F_{222}\! \left(x \right)+F_{224}\! \left(x \right)\\
F_{222}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{225}\! \left(x \right)\\
F_{225}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{212}\! \left(x \right)+F_{226}\! \left(x \right)\\
F_{226}\! \left(x \right) &= F_{227}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{227}\! \left(x \right) &= F_{228}\! \left(x \right)+F_{230}\! \left(x \right)\\
F_{228}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{229}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{236}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{233}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{234}\! \left(x \right)+F_{235}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{223}\! \left(x \right)\\
F_{235}\! \left(x \right) &= F_{231}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{237}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{238}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{238}\! \left(x \right) &= F_{239}\! \left(x \right)+F_{240}\! \left(x \right)\\
F_{239}\! \left(x \right) &= F_{229}\! \left(x \right)\\
F_{240}\! \left(x \right) &= F_{241}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{242}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{243}\! \left(x \right)+F_{244}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{229}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{241}\! \left(x \right)\\
F_{245}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{246}\! \left(x \right)\\
F_{246}\! \left(x \right) &= F_{187}\! \left(x \right)\\
F_{247}\! \left(x \right) &= F_{248}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{248}\! \left(x \right) &= F_{249}\! \left(x \right)+F_{311}\! \left(x \right)\\
F_{249}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{250}\! \left(x \right)\\
F_{250}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{251}\! \left(x \right)+F_{308}\! \left(x \right)\\
F_{251}\! \left(x \right) &= F_{252}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{255}\! \left(x \right)\\
F_{253}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{254}\! \left(x \right)\\
F_{254}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{255}\! \left(x \right) &= F_{256}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{256}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{216}\! \left(x \right)+F_{257}\! \left(x \right)\\
F_{257}\! \left(x \right) &= F_{258}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{258}\! \left(x \right) &= F_{259}\! \left(x \right)+F_{261}\! \left(x \right)\\
F_{259}\! \left(x \right) &= F_{254}\! \left(x \right)+F_{260}\! \left(x \right)\\
F_{260}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{112}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{261}\! \left(x \right) &= F_{262}\! \left(x \right)+F_{288}\! \left(x \right)\\
F_{262}\! \left(x \right) &= F_{263}\! \left(x \right)\\
F_{263}\! \left(x \right) &= F_{264}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{264}\! \left(x \right) &= F_{265}\! \left(x \right)+F_{267}\! \left(x \right)\\
F_{265}\! \left(x \right) &= F_{254}\! \left(x \right)+F_{266}\! \left(x \right)\\
F_{266}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{267}\! \left(x \right) &= F_{262}\! \left(x \right)+F_{268}\! \left(x \right)\\
F_{268}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{156}\! \left(x \right)+F_{269}\! \left(x \right)\\
F_{269}\! \left(x \right) &= F_{270}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{270}\! \left(x \right) &= F_{271}\! \left(x \right)+F_{273}\! \left(x \right)\\
F_{271}\! \left(x \right) &= F_{266}\! \left(x \right)+F_{272}\! \left(x \right)\\
F_{272}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{273}\! \left(x \right) &= F_{274}\! \left(x \right)+F_{279}\! \left(x \right)\\
F_{274}\! \left(x \right) &= F_{275}\! \left(x \right)\\
F_{275}\! \left(x \right) &= F_{276}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{276}\! \left(x \right) &= F_{277}\! \left(x \right)+F_{278}\! \left(x \right)\\
F_{277}\! \left(x \right) &= F_{266}\! \left(x \right)\\
F_{278}\! \left(x \right) &= F_{274}\! \left(x \right)\\
F_{279}\! \left(x \right) &= F_{280}\! \left(x \right)\\
F_{280}\! \left(x \right) &= F_{281}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{281}\! \left(x \right) &= F_{282}\! \left(x \right)+F_{283}\! \left(x \right)\\
F_{282}\! \left(x \right) &= F_{272}\! \left(x \right)\\
F_{283}\! \left(x \right) &= F_{284}\! \left(x \right)\\
F_{284}\! \left(x \right) &= F_{285}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{285}\! \left(x \right) &= F_{286}\! \left(x \right)+F_{287}\! \left(x \right)\\
F_{286}\! \left(x \right) &= F_{272}\! \left(x \right)\\
F_{287}\! \left(x \right) &= F_{284}\! \left(x \right)\\
F_{288}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{212}\! \left(x \right)+F_{289}\! \left(x \right)\\
F_{289}\! \left(x \right) &= F_{290}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{290}\! \left(x \right) &= F_{291}\! \left(x \right)+F_{293}\! \left(x \right)\\
F_{291}\! \left(x \right) &= F_{260}\! \left(x \right)+F_{292}\! \left(x \right)\\
F_{292}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{293}\! \left(x \right) &= F_{294}\! \left(x \right)+F_{299}\! \left(x \right)\\
F_{294}\! \left(x \right) &= F_{295}\! \left(x \right)\\
F_{295}\! \left(x \right) &= F_{296}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{296}\! \left(x \right) &= F_{297}\! \left(x \right)+F_{298}\! \left(x \right)\\
F_{297}\! \left(x \right) &= F_{260}\! \left(x \right)\\
F_{298}\! \left(x \right) &= F_{294}\! \left(x \right)\\
F_{299}\! \left(x \right) &= F_{300}\! \left(x \right)\\
F_{300}\! \left(x \right) &= F_{301}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{301}\! \left(x \right) &= F_{302}\! \left(x \right)+F_{303}\! \left(x \right)\\
F_{302}\! \left(x \right) &= F_{292}\! \left(x \right)\\
F_{303}\! \left(x \right) &= F_{304}\! \left(x \right)\\
F_{304}\! \left(x \right) &= F_{305}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{305}\! \left(x \right) &= F_{306}\! \left(x \right)+F_{307}\! \left(x \right)\\
F_{306}\! \left(x \right) &= F_{292}\! \left(x \right)\\
F_{307}\! \left(x \right) &= F_{304}\! \left(x \right)\\
F_{308}\! \left(x \right) &= F_{309}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{309}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{310}\! \left(x \right)\\
F_{310}\! \left(x \right) &= F_{308}\! \left(x \right)\\
F_{311}\! \left(x \right) &= F_{310}\! \left(x \right)+F_{312}\! \left(x \right)\\
F_{312}\! \left(x \right) &= F_{105}\! \left(x \right)\\
\end{align*}\)