Av(1234, 2431, 3241)
Generating Function
\(\displaystyle -\frac{3 x^{8}+5 x^{7}+13 x^{6}+7 x^{5}+2 x^{4}+x^{3}+5 x^{2}-4 x +1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 231, 650, 1668, 3987, 9030, 19628, 41333, 84915, 171087, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{8}+5 x^{7}+13 x^{6}+7 x^{5}+2 x^{4}+x^{3}+5 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 231\)
\(\displaystyle a \! \left(7\right) = 650\)
\(\displaystyle a \! \left(8\right) = 1668\)
\(\displaystyle a \! \left(n +5\right) = \frac{33 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+\frac{51 n}{2}+33, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 231\)
\(\displaystyle a \! \left(7\right) = 650\)
\(\displaystyle a \! \left(8\right) = 1668\)
\(\displaystyle a \! \left(n +5\right) = \frac{33 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+\frac{51 n}{2}+33, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-8320 \sqrt{11}+32340 \,\mathrm{I}\right) \sqrt{3}-24960 \,\mathrm{I} \sqrt{11}+32340\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+100320+\left(\left(11105 \sqrt{11}+68475 \,\mathrm{I}\right) \sqrt{3}-33315 \,\mathrm{I} \sqrt{11}-68475\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5280}\\+\\\frac{\left(\left(\left(11105 \sqrt{11}-68475 \,\mathrm{I}\right) \sqrt{3}+33315 \,\mathrm{I} \sqrt{11}-68475\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+100320+\left(\left(-8320 \sqrt{11}-32340 \,\mathrm{I}\right) \sqrt{3}+24960 \,\mathrm{I} \sqrt{11}+32340\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5280}\\+\\\frac{\left(\left(-22210 \sqrt{11}\, \sqrt{3}+136950\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+16640 \sqrt{11}\, \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-64680 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+100320\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{5280}\\+\frac{\left(166848 \sqrt{5}-369600\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}+\\\frac{\left(-166848 \sqrt{5}-369600\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}+\frac{33 n^{2}}{4}+\frac{117 n}{4}+87 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 335 rules.
Found on January 18, 2022.Finding the specification took 30 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_{40}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{38}\! \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_{22}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \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_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{239}\! \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_{57}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \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_{13}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \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_{54}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{64}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{76}\! \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_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{85}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{99}\! \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_{107}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{114}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{115}\! \left(x \right)+F_{205}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{118}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{119}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{179}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{129}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{141}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{126}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{145}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{145}\! \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_{149}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{146}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{158}\! \left(x \right)\\
F_{158}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{150}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{163}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{173}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{174}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{179}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{180}\! \left(x \right)+F_{201}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{186}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{184}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{192}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{188}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{189}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)+F_{191}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{187}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{193}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{194}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)+F_{196}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{183}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)+F_{200}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{183}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{197}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{202}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{203}\! \left(x \right)+F_{204}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{153}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{205}\! \left(x \right) &= F_{206}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{233}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)\\
F_{209}\! \left(x \right) &= F_{210}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{212}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{212}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{213}\! \left(x \right)\\
F_{213}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{201}\! \left(x \right)+F_{214}\! \left(x \right)\\
F_{214}\! \left(x \right) &= F_{215}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{215}\! \left(x \right) &= F_{216}\! \left(x \right)+F_{218}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{217}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{184}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{224}\! \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_{223}\! \left(x \right)\\
F_{222}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{219}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{225}\! \left(x \right)\\
F_{225}\! \left(x \right) &= F_{226}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{226}\! \left(x \right) &= F_{227}\! \left(x \right)+F_{228}\! \left(x \right)\\
F_{227}\! \left(x \right) &= F_{217}\! \left(x \right)\\
F_{228}\! \left(x \right) &= F_{229}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{230}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{232}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{217}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{229}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{234}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)\\
F_{235}\! \left(x \right) &= F_{236}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{237}\! \left(x \right)+F_{238}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{238}\! \left(x \right) &= F_{219}\! \left(x \right)\\
F_{239}\! \left(x \right) &= F_{240}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{240}\! \left(x \right) &= F_{241}\! \left(x \right)+F_{301}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{242}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{243}\! \left(x \right)+F_{299}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{244}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{246}\! \left(x \right)\\
F_{245}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{246}\! \left(x \right) &= F_{247}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{247}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{205}\! \left(x \right)+F_{248}\! \left(x \right)\\
F_{248}\! \left(x \right) &= F_{249}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{249}\! \left(x \right) &= F_{250}\! \left(x \right)+F_{252}\! \left(x \right)\\
F_{250}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{251}\! \left(x \right)\\
F_{251}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{119}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{279}\! \left(x \right)\\
F_{253}\! \left(x \right) &= F_{254}\! \left(x \right)\\
F_{254}\! \left(x \right) &= F_{255}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{255}\! \left(x \right) &= F_{256}\! \left(x \right)+F_{258}\! \left(x \right)\\
F_{256}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{257}\! \left(x \right)\\
F_{257}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{258}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{259}\! \left(x \right)\\
F_{259}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{150}\! \left(x \right)+F_{260}\! \left(x \right)\\
F_{260}\! \left(x \right) &= F_{261}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{261}\! \left(x \right) &= F_{262}\! \left(x \right)+F_{264}\! \left(x \right)\\
F_{262}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{263}\! \left(x \right)\\
F_{263}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{264}\! \left(x \right) &= F_{265}\! \left(x \right)+F_{270}\! \left(x \right)\\
F_{265}\! \left(x \right) &= F_{266}\! \left(x \right)\\
F_{266}\! \left(x \right) &= F_{267}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)+F_{269}\! \left(x \right)\\
F_{268}\! \left(x \right) &= F_{257}\! \left(x \right)\\
F_{269}\! \left(x \right) &= F_{265}\! \left(x \right)\\
F_{270}\! \left(x \right) &= F_{271}\! \left(x \right)\\
F_{271}\! \left(x \right) &= F_{272}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{272}\! \left(x \right) &= F_{273}\! \left(x \right)+F_{274}\! \left(x \right)\\
F_{273}\! \left(x \right) &= F_{263}\! \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_{263}\! \left(x \right)\\
F_{278}\! \left(x \right) &= F_{275}\! \left(x \right)\\
F_{279}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{201}\! \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_{284}\! \left(x \right)\\
F_{282}\! \left(x \right) &= F_{251}\! \left(x \right)+F_{283}\! \left(x \right)\\
F_{283}\! \left(x \right) &= F_{184}\! \left(x \right)\\
F_{284}\! \left(x \right) &= F_{285}\! \left(x \right)+F_{290}\! \left(x \right)\\
F_{285}\! \left(x \right) &= F_{286}\! \left(x \right)\\
F_{286}\! \left(x \right) &= F_{287}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{287}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{289}\! \left(x \right)\\
F_{288}\! \left(x \right) &= F_{251}\! \left(x \right)\\
F_{289}\! \left(x \right) &= F_{285}\! \left(x \right)\\
F_{290}\! \left(x \right) &= F_{291}\! \left(x \right)\\
F_{291}\! \left(x \right) &= F_{292}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{292}\! \left(x \right) &= F_{293}\! \left(x \right)+F_{294}\! \left(x \right)\\
F_{293}\! \left(x \right) &= F_{283}\! \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_{283}\! \left(x \right)\\
F_{298}\! \left(x \right) &= F_{295}\! \left(x \right)\\
F_{299}\! \left(x \right) &= F_{300}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{300}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{301}\! \left(x \right) &= F_{302}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{302}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{303}\! \left(x \right)+F_{333}\! \left(x \right)\\
F_{303}\! \left(x \right) &= F_{304}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{304}\! \left(x \right) &= F_{305}\! \left(x \right)+F_{306}\! \left(x \right)\\
F_{305}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{306}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{307}\! \left(x \right)\\
F_{307}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{308}\! \left(x \right)+F_{329}\! \left(x \right)\\
F_{308}\! \left(x \right) &= F_{309}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{309}\! \left(x \right) &= F_{310}\! \left(x \right)+F_{314}\! \left(x \right)\\
F_{310}\! \left(x \right) &= F_{311}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{311}\! \left(x \right) &= F_{312}\! \left(x \right)\\
F_{312}\! \left(x \right) &= F_{313}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{313}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{314}\! \left(x \right) &= F_{315}\! \left(x \right)+F_{320}\! \left(x \right)\\
F_{315}\! \left(x \right) &= F_{316}\! \left(x \right)\\
F_{316}\! \left(x \right) &= F_{317}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{317}\! \left(x \right) &= F_{318}\! \left(x \right)+F_{319}\! \left(x \right)\\
F_{318}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{319}\! \left(x \right) &= F_{315}\! \left(x \right)\\
F_{320}\! \left(x \right) &= F_{321}\! \left(x \right)\\
F_{321}\! \left(x \right) &= F_{322}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{322}\! \left(x \right) &= F_{323}\! \left(x \right)+F_{324}\! \left(x \right)\\
F_{323}\! \left(x \right) &= F_{311}\! \left(x \right)\\
F_{324}\! \left(x \right) &= F_{325}\! \left(x \right)\\
F_{325}\! \left(x \right) &= F_{326}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)+F_{328}\! \left(x \right)\\
F_{327}\! \left(x \right) &= F_{311}\! \left(x \right)\\
F_{328}\! \left(x \right) &= F_{325}\! \left(x \right)\\
F_{329}\! \left(x \right) &= F_{330}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{330}\! \left(x \right) &= F_{331}\! \left(x \right)+F_{332}\! \left(x \right)\\
F_{331}\! \left(x \right) &= F_{153}\! \left(x \right)\\
F_{332}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{333}\! \left(x \right) &= F_{334}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{334}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{99}\! \left(x \right)\\
\end{align*}\)