Av(1234, 1432, 2341, 2413)
Generating Function
\(\displaystyle -\frac{2 x^{8}-2 x^{7}+2 x^{6}+5 x^{5}-6 x^{4}+7 x^{3}-9 x^{2}+5 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 59, 159, 416, 1075, 2752, 6987, 17613, 44130, 109993, 272913, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+2 x^{8}-2 x^{7}+2 x^{6}+5 x^{5}-6 x^{4}+7 x^{3}-9 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 416\)
\(\displaystyle a \! \left(8\right) = 1075\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 416\)
\(\displaystyle a \! \left(8\right) = 1075\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(-345 \left(\left(\left(\frac{n}{3}+\frac{4486}{345}\right) \sqrt{23}+\mathrm{I} n -\frac{1184 \,\mathrm{I}}{15}\right) \sqrt{3}+\left(\mathrm{I} n +\frac{4486 \,\mathrm{I}}{115}\right) \sqrt{23}+n -\frac{1184}{15}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-2760 \left(\left(\left(\frac{n}{12}-\frac{647}{552}\right) \sqrt{23}+\mathrm{I} n +\frac{547 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}+\frac{647 \,\mathrm{I}}{184}\right) \sqrt{23}-n -\frac{547}{24}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+13800 n +740600\right) \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(2760 \left(\left(\left(-\frac{n}{12}+\frac{647}{552}\right) \sqrt{23}+\mathrm{I} n +\frac{547 \,\mathrm{I}}{24}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{4}+\frac{647 \,\mathrm{I}}{184}\right) \sqrt{23}+n +\frac{547}{24}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+345 \left(\left(\left(-\frac{n}{3}-\frac{4486}{345}\right) \sqrt{23}+\mathrm{I} n -\frac{1184 \,\mathrm{I}}{15}\right) \sqrt{3}+\left(\mathrm{I} n +\frac{4486 \,\mathrm{I}}{115}\right) \sqrt{23}-n +\frac{1184}{15}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+13800 n +740600\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n} \left(2 \left(\sqrt{3}\, \left(n -\frac{647}{46}\right) \sqrt{23}-12 n -\frac{547}{2}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\sqrt{3}\, \left(n +\frac{4486}{115}\right) \sqrt{23}+3 n -\frac{1184}{5}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+60 n +3220\right)}{1380} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 176 rules.
Found on January 18, 2022.Finding the specification took 6 seconds.
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Copy 176 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_{26}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{18}\! \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_{46}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{40}\! \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_{36}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{17}\! \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_{35}\! \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_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{43}\! \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_{11}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{166}\! \left(x \right)+F_{18}\! \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_{50}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)+F_{62}\! \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_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 0\\
F_{63}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{122}\! \left(x \right)+F_{158}\! \left(x \right)+F_{18}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)+F_{18}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 0\\
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_{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_{83}\! \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_{92}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{109}\! \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_{100}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{101}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{102}\! \left(x \right)+F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{106}\! \left(x \right) &= 0\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{80}\! \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_{83}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{113}\! \left(x \right) &= 0\\
F_{114}\! \left(x \right) &= 0\\
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_{118}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \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_{83}\! \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_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \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_{137}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{11}\! \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_{135}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{139}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{145}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{142}\! \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_{43}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{147}\! \left(x \right)+F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{146}\! \left(x \right)\\
F_{151}\! \left(x \right) &= 0\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{125}\! \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_{128}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{158}\! \left(x \right) &= 0\\
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_{162}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{163}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{128}\! \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_{168}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{169}\! \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_{2}\! \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_{128}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{169}\! \left(x \right)\\
\end{align*}\)