Av(1243, 1342, 1432, 3124, 3142)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)}{x^{4}+2 x^{3}-4 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 166, 480, 1389, 4025, 11670, 33838, 98111, 284457, 824730, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle -\frac{\left(-352080000 \left(\frac{\sqrt{120 \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(18 \sqrt{163}\, \sqrt{3}-294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+26400}}{120}+\frac{\left(-49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-2200\right) \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}+\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\left(\left(\left(-\frac{62 \sqrt{163}}{5}+\frac{2282 \sqrt{3}}{5}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(1630 \sqrt{3}-350 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-310 \sqrt{163}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(2691 \sqrt{163}\, \sqrt{3}+60147\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-31230 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-70200 \sqrt{163}\, \sqrt{3}+689490 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(-7172 \sqrt{3}+1152 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-21190 \sqrt{3}-810 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-130400 \sqrt{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(40860 \sqrt{163}\, \sqrt{3}-205380\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+138600 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-4987800 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-363816000\right) \left(-\frac{\sqrt{120 \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(18 \sqrt{163}\, \sqrt{3}-294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+26400}}{120}+\frac{\left(-49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-2200\right) \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}+\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\left(\left(\left(\left(\frac{571 \,\mathrm{I} \sqrt{3}}{40}+\frac{1421}{120}\right) \sqrt{163}-\frac{28379 \,\mathrm{I}}{120}+\frac{18419 \sqrt{3}}{120}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\mathrm{I} \sqrt{3}-\frac{1177}{6}\right) \sqrt{163}-\frac{2806 \,\mathrm{I}}{3}+\frac{7987 \sqrt{3}}{6}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{30065 \,\mathrm{I}}{3}-\frac{1585 \sqrt{163}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(414 \,\mathrm{I}+2901 \sqrt{3}\right) \sqrt{163}-2604 \,\mathrm{I} \sqrt{3}+1467\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-315 \,\mathrm{I}-14655 \sqrt{3}\right) \sqrt{163}-7485 \,\mathrm{I} \sqrt{3}+432765\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-740100 \left(\mathrm{I}+\frac{215 \sqrt{163}}{2467}\right) \sqrt{3}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(\left(\frac{331 \,\mathrm{I} \sqrt{3}}{3}-1228\right) \sqrt{163}-1141 \,\mathrm{I}+\frac{15974 \sqrt{3}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{595 \,\mathrm{I} \sqrt{3}}{3}-1225\right) \sqrt{163}+10595 \,\mathrm{I}+\frac{101875 \sqrt{3}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1400 \sqrt{3}\, \left(\mathrm{I} \sqrt{163}+\frac{7661}{7}\right)}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(61290 \,\mathrm{I}-20430 \sqrt{3}\right) \sqrt{163}-102690 \,\mathrm{I} \sqrt{3}+102690\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-363816000+\left(\left(-207900 \,\mathrm{I}-69300 \sqrt{3}\right) \sqrt{163}+2493900 \,\mathrm{I} \sqrt{3}+2493900\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(-\frac{\mathrm{I} \sqrt{120 \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(-18 \sqrt{163}\, \sqrt{3}+294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}-26400}}{120}+\frac{\left(49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+2200\right) \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}-\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\frac{\mathrm{I} \sqrt{120 \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(-18 \sqrt{163}\, \sqrt{3}+294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}-26400}}{120}+\frac{\left(49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+2200\right) \sqrt{\left(120 \sqrt{163}\, 2^{\frac{1}{3}} \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}-\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n} \left(\left(\left(\left(\left(-\frac{571 \,\mathrm{I} \sqrt{3}}{40}+\frac{1421}{120}\right) \sqrt{163}+\frac{28379 \,\mathrm{I}}{120}+\frac{18419 \sqrt{3}}{120}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\mathrm{I} \sqrt{3}-\frac{1177}{6}\right) \sqrt{163}+\frac{2806 \,\mathrm{I}}{3}+\frac{7987 \sqrt{3}}{6}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{30065 \,\mathrm{I}}{3}-\frac{1585 \sqrt{163}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(-414 \,\mathrm{I}+2901 \sqrt{3}\right) \sqrt{163}+2604 \,\mathrm{I} \sqrt{3}+1467\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(315 \,\mathrm{I}-14655 \sqrt{3}\right) \sqrt{163}+7485 \,\mathrm{I} \sqrt{3}+432765\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+740100 \left(\mathrm{I}-\frac{215 \sqrt{163}}{2467}\right) \sqrt{3}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(\left(-\frac{331 \,\mathrm{I} \sqrt{3}}{3}-1228\right) \sqrt{163}+1141 \,\mathrm{I}+\frac{15974 \sqrt{3}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{595 \,\mathrm{I} \sqrt{3}}{3}-1225\right) \sqrt{163}-10595 \,\mathrm{I}+\frac{101875 \sqrt{3}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{1400 \left(\mathrm{I} \sqrt{163}-\frac{7661}{7}\right) \sqrt{3}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(-61290 \,\mathrm{I}-20430 \sqrt{3}\right) \sqrt{163}+102690 \,\mathrm{I} \sqrt{3}+102690\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-363816000+\left(\left(207900 \,\mathrm{I}-69300 \sqrt{3}\right) \sqrt{163}-2493900 \,\mathrm{I} \sqrt{3}+2493900\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right)\right) \left(\left(\left(\left(\sqrt{3}-\frac{43 \sqrt{163}}{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\frac{2 \sqrt{3}}{5}-\frac{16 \sqrt{163}}{815}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{20 \sqrt{163}}{163}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}-\frac{4140 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}}{163}+\frac{306 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}}{163}-\frac{7200 \sqrt{163}\, \sqrt{3}}{163}+540 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+54 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+648000+\left(\left(40 \sqrt{3}-\frac{120 \sqrt{163}}{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(8 \sqrt{3}-\frac{264 \sqrt{163}}{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-400 \sqrt{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}\right)}{912591360000000}\)
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This specification was found using the strategy pack "Point Placements" and has 37 rules.

Found on January 18, 2022.

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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_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{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_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \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_{33}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \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_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)\\ \end{align*}\)