Av(13452, 14352, 31452, 34152, 41352, 43152)
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Counting Sequence
1, 1, 2, 6, 24, 114, 600, 3366, 19704, 118890, 733836, 4610514, 29382564, 189463698, 1233776172, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(-3 x +2\right) F \left(x \right)^{3}+\left(6 x^{3}-2 x^{2}+11 x -7\right) F \left(x \right)^{2}+\left(2 x^{4}-x^{3}-8 x^{2}-11 x +8\right) F \! \left(x \right)+3 \left(x +1\right) \left(x^{2}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a(6) = 600\)
\(\displaystyle a(7) = 3366\)
\(\displaystyle a(8) = 19704\)
\(\displaystyle a(9) = 118890\)
\(\displaystyle a(10) = 733836\)
\(\displaystyle a(11) = 4610514\)
\(\displaystyle a(12) = 29382564\)
\(\displaystyle a(13) = 189463698\)
\(\displaystyle a(14) = 1233776172\)
\(\displaystyle a(15) = 8101888338\)
\(\displaystyle a(16) = 53588885928\)
\(\displaystyle a(17) = 356696620974\)
\(\displaystyle a(18) = 2387426484576\)
\(\displaystyle a{\left(n + 19 \right)} = - \frac{3 \left(n - 2\right) \left(n - 1\right) a{\left(n \right)}}{\left(n + 17\right) \left(n + 19\right)} + \frac{\left(n - 1\right) \left(25 n + 4\right) a{\left(n + 1 \right)}}{2 \left(n + 17\right) \left(n + 19\right)} - \frac{3 \left(103 n^{2} - 611 n - 946\right) a{\left(n + 3 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(133 n^{2} + 4539 n + 38636\right) a{\left(n + 18 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(881 n^{2} + 1965 n + 52\right) a{\left(n + 2 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(3923 n^{2} + 126627 n + 1020082\right) a{\left(n + 17 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} - \frac{3 \left(18039 n^{2} + 155809 n + 382624\right) a{\left(n + 6 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(32441 n^{2} + 987339 n + 7503658\right) a{\left(n + 16 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(39317 n^{2} + 250521 n + 340678\right) a{\left(n + 4 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(53533 n^{2} + 440622 n + 847553\right) a{\left(n + 5 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(78593 n^{2} + 2249898 n + 16092175\right) a{\left(n + 15 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(105335 n^{2} + 1257150 n + 3573277\right) a{\left(n + 7 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(114821 n^{2} + 828021 n - 755486\right) a{\left(n + 9 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(214913 n^{2} + 5845476 n + 39369805\right) a{\left(n + 13 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(420973 n^{2} + 11389173 n + 77003036\right) a{\left(n + 14 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(519949 n^{2} + 6661017 n + 21756170\right) a{\left(n + 8 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(627745 n^{2} + 11089899 n + 45562286\right) a{\left(n + 12 \right)}}{8 \left(n + 17\right) \left(n + 19\right)} - \frac{\left(915742 n^{2} + 16112157 n + 71041430\right) a{\left(n + 10 \right)}}{4 \left(n + 17\right) \left(n + 19\right)} + \frac{\left(2172833 n^{2} + 40612287 n + 188788006\right) a{\left(n + 11 \right)}}{8 \left(n + 17\right) \left(n + 19\right)}, \quad n \geq 19\)
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This specification was found using the strategy pack "Point Placements Req Corrob" and has 45 rules.

Finding the specification took 5869 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{10}\! \left(x \right) F_{4}\! \left(x \right)}\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= -F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{25}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{5}\! \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_{12}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right) F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{35}\! \left(x \right) &= -F_{15}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{37}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{17}\! \left(x \right) F_{39}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 133 rules.

Finding the specification took 17174 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_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{12}\! \left(x \right) x +F_{12} \left(x \right)^{2}+x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{19}\! \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_{53}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{34}\! \left(x \right) x +F_{34} \left(x \right)^{2}-2 F_{34}\! \left(x \right)+2\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right) F_{38}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= -F_{52}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= -F_{51}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{50}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= -F_{29}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{57}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{2}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{2}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{27}\! \left(x \right) F_{38}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{2}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{27}\! \left(x \right) F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{2}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{42}\! \left(x \right) F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{2}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{38}\! \left(x \right) F_{74}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{2}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{53}\! \left(x \right) F_{74}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{12}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{35}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{2}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{106}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{117}\! \left(x \right) &= -F_{25}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{115}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right) F_{8}\! \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_{125}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{123}\! \left(x \right)\\ \end{align*}\)