Av(13425, 14235, 14325, 24153, 31524, 426153)
Counting Sequence
1, 1, 2, 6, 24, 115, 607, 3370, 19235, 111571, 653603, 3853424, 22815443, 135482334, 806180411, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(13 x^{9}-69 x^{8}+162 x^{7}-115 x^{6}-496 x^{5}+1401 x^{4}-1415 x^{3}+637 x^{2}-97 x +4\right) F \left(x
\right)^{3}+\left(50 x^{8}-198 x^{7}+376 x^{6}+118 x^{5}-1794 x^{4}+2840 x^{3}-1676 x^{2}+281 x -12\right) F \left(x
\right)^{2}+\left(60 x^{7}-192 x^{6}+248 x^{5}+483 x^{4}-1743 x^{3}+1472 x^{2}-272 x +12\right) F \! \left(x \right)+24 x^{6}-64 x^{5}+40 x^{4}+295 x^{3}-432 x^{2}+88 x -4 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 115\)
\(\displaystyle a \! \left(6\right) = 607\)
\(\displaystyle a \! \left(7\right) = 3370\)
\(\displaystyle a \! \left(8\right) = 19235\)
\(\displaystyle a \! \left(9\right) = 111571\)
\(\displaystyle a \! \left(10\right) = 653603\)
\(\displaystyle a \! \left(11\right) = 3853424\)
\(\displaystyle a \! \left(12\right) = 22815443\)
\(\displaystyle a \! \left(13\right) = 135482334\)
\(\displaystyle a \! \left(14\right) = 806180411\)
\(\displaystyle a \! \left(15\right) = 4804250596\)
\(\displaystyle a \! \left(16\right) = 28660891938\)
\(\displaystyle a \! \left(17\right) = 171120790746\)
\(\displaystyle a \! \left(18\right) = 1022298997790\)
\(\displaystyle a \! \left(19\right) = 6110152136891\)
\(\displaystyle a \! \left(20\right) = 36532429252431\)
\(\displaystyle a \! \left(21\right) = 218485628254633\)
\(\displaystyle a \! \left(22\right) = 1306949786188459\)
\(\displaystyle a \! \left(23\right) = 7819280602197891\)
\(\displaystyle a \! \left(24\right) = 46787663410514384\)
\(\displaystyle a \! \left(25\right) = 279988909890506863\)
\(\displaystyle a \! \left(26\right) = 1675660680553158236\)
\(\displaystyle a \! \left(n +27\right) = -\frac{3 \left(2870 n^{2}+143228 n +1786491\right) a \! \left(n +25\right)}{4 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(150 n^{2}+7639 n +97258\right) a \! \left(n +26\right)}{4 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(24246610792 n^{2}+858496776663 n +7605196753667\right) a \! \left(n +18\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(12483090055 n^{2}+460618546182 n +4250174677163\right) a \! \left(n +19\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(2059267847 n^{2}+79071121917 n +758781691441\right) a \! \left(n +20\right)}{8 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(141216301 n^{2}+5631113645 n +56078045437\right) a \! \left(n +21\right)}{8 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(51035902 n^{2}+2101523442 n +21584384003\right) a \! \left(n +22\right)}{8 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(2842105 n^{2}+117679806 n +1208358899\right) a \! \left(n +23\right)}{8 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(9178 n^{2}+532642 n +7453545\right) a \! \left(n +24\right)}{4 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(30721703953 n^{2}+1040895182118 n +8828090853959\right) a \! \left(n +17\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(5234940499 n^{2}+131947290891 n +820854701750\right) a \! \left(n +14\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(9580485722 n^{2}+302061226695 n +2378158241950\right) a \! \left(n +15\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(24502373768 n^{2}+793768381311 n +6436861822186\right) a \! \left(n +16\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(12072591 n^{2}+180302185 n +664492260\right) a \! \left(n +6\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(168501335 n^{2}+2504088123 n +9161071858\right) a \! \left(n +7\right)}{32 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(140883621 n^{2}+2177583625 n +8298517096\right) a \! \left(n +8\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(529384081 n^{2}+8650037967 n +35244539990\right) a \! \left(n +9\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(104225549 n^{2}+2780641806 n +17171348851\right) a \! \left(n +10\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(427330215 n^{2}+10124230856 n +60248773028\right) a \! \left(n +11\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(3683866163 n^{2}+93617955987 n +596768869546\right) a \! \left(n +12\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(3935950077 n^{2}+104390310173 n +693195213084\right) a \! \left(n +13\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(458405 n^{2}+2637517 n +3726348\right) a \! \left(n +2\right)}{32 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(3773657 n^{2}+24614913 n +38867068\right) a \! \left(n +3\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(7368883 n^{2}+60653781 n +129990848\right) a \! \left(n +4\right)}{32 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(2179541 n^{2}+14707425 n +16019218\right) a \! \left(n +5\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{7917 \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{8 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(75847 n +193159\right) \left(n +2\right) a \! \left(n +1\right)}{8 \left(2 n +53\right) \left(n +26\right)}, \quad n \geq 27\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 115\)
\(\displaystyle a \! \left(6\right) = 607\)
\(\displaystyle a \! \left(7\right) = 3370\)
\(\displaystyle a \! \left(8\right) = 19235\)
\(\displaystyle a \! \left(9\right) = 111571\)
\(\displaystyle a \! \left(10\right) = 653603\)
\(\displaystyle a \! \left(11\right) = 3853424\)
\(\displaystyle a \! \left(12\right) = 22815443\)
\(\displaystyle a \! \left(13\right) = 135482334\)
\(\displaystyle a \! \left(14\right) = 806180411\)
\(\displaystyle a \! \left(15\right) = 4804250596\)
\(\displaystyle a \! \left(16\right) = 28660891938\)
\(\displaystyle a \! \left(17\right) = 171120790746\)
\(\displaystyle a \! \left(18\right) = 1022298997790\)
\(\displaystyle a \! \left(19\right) = 6110152136891\)
\(\displaystyle a \! \left(20\right) = 36532429252431\)
\(\displaystyle a \! \left(21\right) = 218485628254633\)
\(\displaystyle a \! \left(22\right) = 1306949786188459\)
\(\displaystyle a \! \left(23\right) = 7819280602197891\)
\(\displaystyle a \! \left(24\right) = 46787663410514384\)
\(\displaystyle a \! \left(25\right) = 279988909890506863\)
\(\displaystyle a \! \left(26\right) = 1675660680553158236\)
\(\displaystyle a \! \left(n +27\right) = -\frac{3 \left(2870 n^{2}+143228 n +1786491\right) a \! \left(n +25\right)}{4 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(150 n^{2}+7639 n +97258\right) a \! \left(n +26\right)}{4 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(24246610792 n^{2}+858496776663 n +7605196753667\right) a \! \left(n +18\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(12483090055 n^{2}+460618546182 n +4250174677163\right) a \! \left(n +19\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(2059267847 n^{2}+79071121917 n +758781691441\right) a \! \left(n +20\right)}{8 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(141216301 n^{2}+5631113645 n +56078045437\right) a \! \left(n +21\right)}{8 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(51035902 n^{2}+2101523442 n +21584384003\right) a \! \left(n +22\right)}{8 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(2842105 n^{2}+117679806 n +1208358899\right) a \! \left(n +23\right)}{8 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(9178 n^{2}+532642 n +7453545\right) a \! \left(n +24\right)}{4 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(30721703953 n^{2}+1040895182118 n +8828090853959\right) a \! \left(n +17\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(5234940499 n^{2}+131947290891 n +820854701750\right) a \! \left(n +14\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(9580485722 n^{2}+302061226695 n +2378158241950\right) a \! \left(n +15\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(24502373768 n^{2}+793768381311 n +6436861822186\right) a \! \left(n +16\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(12072591 n^{2}+180302185 n +664492260\right) a \! \left(n +6\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(168501335 n^{2}+2504088123 n +9161071858\right) a \! \left(n +7\right)}{32 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(140883621 n^{2}+2177583625 n +8298517096\right) a \! \left(n +8\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(529384081 n^{2}+8650037967 n +35244539990\right) a \! \left(n +9\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(104225549 n^{2}+2780641806 n +17171348851\right) a \! \left(n +10\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(427330215 n^{2}+10124230856 n +60248773028\right) a \! \left(n +11\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(3683866163 n^{2}+93617955987 n +596768869546\right) a \! \left(n +12\right)}{16 \left(2 n +53\right) \left(n +26\right)}-\frac{3 \left(3935950077 n^{2}+104390310173 n +693195213084\right) a \! \left(n +13\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{3 \left(458405 n^{2}+2637517 n +3726348\right) a \! \left(n +2\right)}{32 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(3773657 n^{2}+24614913 n +38867068\right) a \! \left(n +3\right)}{32 \left(2 n +53\right) \left(n +26\right)}+\frac{\left(7368883 n^{2}+60653781 n +129990848\right) a \! \left(n +4\right)}{32 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(2179541 n^{2}+14707425 n +16019218\right) a \! \left(n +5\right)}{16 \left(2 n +53\right) \left(n +26\right)}+\frac{7917 \left(n +2\right) \left(n +1\right) a \! \left(n \right)}{8 \left(2 n +53\right) \left(n +26\right)}-\frac{\left(75847 n +193159\right) \left(n +2\right) a \! \left(n +1\right)}{8 \left(2 n +53\right) \left(n +26\right)}, \quad n \geq 27\)
Heatmap
To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).
This specification was found using the strategy pack "Point Placements" and has 163 rules.
Found on November 07, 2021.Finding the specification took 266100 seconds.
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Copy 163 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_{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_{34}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \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_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \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_{11} \left(x \right)^{2} F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right) F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{0}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{11}\! \left(x \right) F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{1}\! \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_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49} \left(x \right)^{2} F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right) F_{49}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{65}\! \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_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{17}\! \left(x \right) F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{76}\! \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_{103}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{0}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{48}\! \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_{48}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{48}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{10}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{11}\! \left(x \right) F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{10}\! \left(x \right) F_{113}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{48}\! \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_{116}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right) F_{49}\! \left(x \right) F_{52}\! \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_{138}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{46}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{108}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \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_{155}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{11}\! \left(x \right) F_{152}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{11}\! \left(x \right) F_{149}\! \left(x \right) F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \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_{161}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{148}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Untracked Fusion" and has 173 rules.
Found on January 18, 2022.Finding the specification took 271854 seconds.
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Copy 173 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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{8} \left(x \right)^{2} F_{14}\! \left(x \right) F_{5}\! \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_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{2}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{14}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{14}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{34}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{48}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{0}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{14}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right) F_{52}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{14}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{14}\! \left(x \right) F_{51}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{52}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{0}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{14}\! \left(x \right) F_{79}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{52}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{5}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{14}\! \left(x \right) F_{83}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{52}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{14}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{14}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{51}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{26}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{112}\! \left(x \right) F_{14}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{104}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{100}\! \left(x \right) F_{14}\! \left(x \right) F_{52}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{14}\! \left(x \right) F_{52}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{14}\! \left(x \right) F_{55}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{87}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{112}\! \left(x \right) F_{14}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{14}\! \left(x \right) F_{71}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{14}\! \left(x \right) F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{14}\! \left(x \right) F_{145}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{14}\! \left(x \right) F_{148}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{14}\! \left(x \right) F_{48}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{158}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{14}\! \left(x \right) F_{155}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{14}\! \left(x \right) F_{152}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{14}\! \left(x \right) F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{151}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{112}\! \left(x \right) F_{14}\! \left(x \right) F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{14}\! \left(x \right) F_{167}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{14}\! \left(x \right) F_{167}\! \left(x \right) F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{52}\! \left(x \right) F_{53}\! \left(x \right)\\
\end{align*}\)