Av(1234, 1324, 1342, 1423, 3241)
Generating Function
\(\displaystyle \frac{x^{8}-x^{7}-x^{6}-x^{4}+5 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}+x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 155, 406, 1017, 2463, 5818, 13493, 30874, 69951, 157348, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-x^{8}+x^{7}+x^{6}+x^{4}-5 x^{3}+9 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 155\)
\(\displaystyle a \! \left(7\right) = 406\)
\(\displaystyle a \! \left(8\right) = 1017\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)+\left(n +4\right) \left(n +2\right), \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 155\)
\(\displaystyle a \! \left(7\right) = 406\)
\(\displaystyle a \! \left(8\right) = 1017\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+3 a \! \left(n +4\right)+\left(n +4\right) \left(n +2\right), \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{35 \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(-\left(-\frac{2832 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} 576^{n} \left(\left(\frac{47 \sqrt{5}}{10}-\frac{21}{2}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+n^{2}+7 n +16\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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}}{35}+\frac{66552 \left(\sqrt{5}+\frac{105}{47}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\sqrt{5}-1\right)^{-n} 576^{n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{6}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{48}\right)^{n}}{175}-\frac{59 \left(\mathrm{I}-\frac{113 \sqrt{59}}{177}\right) 3^{2 n +\frac{1}{2}} 64^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{35}-\frac{767 \left(\mathrm{I}+\frac{35 \sqrt{59}}{767}\right) 3^{2 n +\frac{1}{2}} 64^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{210}+\left(3^{2 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{767}{105}\right) 2^{6 n +\frac{1}{3}}+\frac{113 \left(\mathrm{I} \sqrt{59}-\frac{59}{113}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{2 n +\frac{1}{3}} 2^{6 n +\frac{2}{3}}}{35}-\frac{944 \,576^{n}}{7}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}\right) \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}}+\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(-\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} \,3^{\frac{1}{6}} 2^{\frac{1}{3}}+\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(\frac{226 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{48}+\frac{3 \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{16}-\mathrm{I} \sqrt{3}-1\right)^{-n} \left(8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}+3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(64^{n} \sqrt{59}\, 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{35 \,64^{n} \sqrt{59}\, 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{226}+\frac{767 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(3^{2 n +\frac{2}{3}} 2^{6 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{3 \,3^{2 n +\frac{1}{3}} 2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}\right)}{113}\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}}{105}+\left(\frac{944 \,9^{n} 32^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\sqrt{59}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}+3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \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}}{7}-\frac{59 \left(\frac{113 \sqrt{59}}{177}+\mathrm{I}\right) 3^{2 n +\frac{1}{2}} 64^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{35}-\frac{767 \left(-\frac{35 \sqrt{59}}{767}+\mathrm{I}\right) 3^{2 n +\frac{1}{2}} 64^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{210}+\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{767}{105}\right) 2^{6 n +\frac{1}{3}}+\frac{113 \left(\mathrm{I} \sqrt{59}+\frac{59}{113}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{2 n +\frac{1}{3}} 2^{6 n +\frac{2}{3}}}{35}+\frac{944 \,576^{n}}{7}\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{\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}\right)\right)}{5664} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 177 rules.
Found on January 18, 2022.Finding the specification took 6 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 177 equations to clipboard:
\(\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_{14}\! \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_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
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_{30}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{65}\! \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_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{7}\! \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_{11}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{40}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{35}\! \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_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{48}\! \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_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{54}\! \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_{54}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{118}\! \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_{72}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{69}\! \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_{24}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{20}\! \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_{24}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{82}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{86}\! \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_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{90}\! \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_{96}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{108}\! \left(x \right)+F_{116}\! \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_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \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_{69}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{127}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{116}\! \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_{125}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{176}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{16}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{145}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{118}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{166}\! \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_{156}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{158}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{155}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{155}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{166}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{116}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{149}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{149}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{139}\! \left(x \right)\\
\end{align*}\)