Av(1342, 1432, 2143, 3142, 3241)
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Generating Function
\(\displaystyle -\frac{3 x^{3}-5 x^{2}+4 x -1}{2 x^{4}-7 x^{3}+8 x^{2}-5 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 59, 181, 554, 1697, 5202, 15950, 48905, 149945, 459731, 1409530, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+3 x^{3}-5 x^{2}+4 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \frac{623819 \left(\frac{3974209810560 \left(\frac{\sqrt{102 \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(18 \sqrt{751}\, \sqrt{3}-150\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}-3468 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+32946}}{408}+\frac{\left(50 \,2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+1156 \left(100+12 \sqrt{751}\, \sqrt{3}\right)^{\frac{1}{3}}+5491\right) \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}}{19455480}-\frac{\sqrt{\left(204272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-7200588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+104829837+\left(28538 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2077266 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{1080860}+\frac{7}{8}\right)^{-n}}{21511}+\left(\left(\left(-\frac{1085946 \left(\frac{23453 \sqrt{751}}{542973}+\sqrt{3}\right) 2^{\frac{5}{6}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}-\frac{7813404 \left(\sqrt{3}-\frac{557 \sqrt{751}}{6759}\right) 2^{\frac{1}{6}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}+\frac{67048 \sqrt{751}\, \sqrt{2}}{439}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}-\frac{18802850 \left(\sqrt{751}\, \sqrt{3}+\frac{29289}{2011}\right) 2^{\frac{5}{6}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}+\frac{75978100 \,2^{\frac{1}{6}} \left(\sqrt{751}\, \sqrt{3}-\frac{24783}{239}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}+\frac{853879400 \sqrt{2}\, \sqrt{751}\, \sqrt{3}}{21511}\right) \sqrt{17 \sqrt{3}\, \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(-3 \sqrt{751}\, \sqrt{3}+25\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+578 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}-5491}+\left(\frac{510680 \left(\sqrt{3}-\frac{11583 \sqrt{751}}{36799}\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{439}+\frac{503530480 \,2^{\frac{2}{3}} \left(\frac{36 \sqrt{751}}{21779}+\sqrt{3}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}+\frac{1180692160 \sqrt{3}}{21511}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}-\frac{572601480 \,2^{\frac{1}{3}} \left(\sqrt{751}\, \sqrt{3}+\frac{80357}{1501}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}+\frac{6623683017600}{21511}-\frac{3125847120 \,2^{\frac{2}{3}} \left(\sqrt{751}\, \sqrt{3}-\frac{10514}{241}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}\right) \left(-\frac{\sqrt{102 \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(18 \sqrt{751}\, \sqrt{3}-150\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}-3468 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+32946}}{408}+\frac{\left(50 \,2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+1156 \left(100+12 \sqrt{751}\, \sqrt{3}\right)^{\frac{1}{3}}+5491\right) \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}}{19455480}-\frac{\sqrt{\left(204272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-7200588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+104829837+\left(28538 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2077266 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{1080860}+\frac{7}{8}\right)^{-n}+\left(\left(\left(-2^{\frac{5}{6}} \left(\left(-\frac{447}{21511}+\mathrm{I} \sqrt{3}\right) \sqrt{751}+\frac{92373 \sqrt{3}}{21511}-\frac{184075 \,\mathrm{I}}{21511}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+\frac{850 \left(\left(\mathrm{I} \sqrt{3}+\frac{1419}{25}\right) \sqrt{751}+\frac{243583 \,\mathrm{I}}{25}-\frac{2253 \sqrt{3}}{25}\right) 2^{\frac{1}{6}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}+\frac{26144096 \sqrt{2}\, \left(\frac{423 \sqrt{751}}{11308}+\mathrm{I}\right)}{21511}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\frac{13337775 \,2^{\frac{5}{6}} \left(\left(-\frac{91 \,\mathrm{I}}{317}-\frac{1559 \sqrt{3}}{2853}\right) \sqrt{751}+\mathrm{I} \sqrt{3}-\frac{5257}{317}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}+\frac{487817550 \,2^{\frac{1}{6}} \left(\left(\frac{\mathrm{I}}{341}+\frac{313 \sqrt{3}}{3069}\right) \sqrt{751}+\mathrm{I} \sqrt{3}-\frac{2253}{341}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}-\frac{8690114400 \sqrt{2}\, \sqrt{3}\, \left(-\frac{13 \sqrt{751}}{804}+\mathrm{I}\right)}{21511}\right) \sqrt{17 \sqrt{3}\, \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(-3 \sqrt{751}\, \sqrt{3}+25\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+578 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}-5491}+\left(\frac{727260 \left(\left(\mathrm{I} \sqrt{3}-\frac{645}{713}\right) \sqrt{751}+\frac{50317 \,\mathrm{I}}{713}+\frac{751 \sqrt{3}}{2139}\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}-\frac{5063280 \,2^{\frac{2}{3}} \left(\left(\mathrm{I} \sqrt{3}+\frac{6}{73}\right) \sqrt{751}-\frac{5257 \,\mathrm{I}}{146}-\frac{3755 \sqrt{3}}{438}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}-\frac{2358240 \left(\mathrm{I} \sqrt{751}+\frac{8261}{3}\right) \sqrt{3}}{21511}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}-\frac{15327294180 \left(\left(\frac{4503 \,\mathrm{I}}{80357}-\frac{1501 \sqrt{3}}{80357}\right) \sqrt{751}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}+\frac{6623683017600}{21511}-\frac{9740710320 \,2^{\frac{2}{3}} \left(\left(-\frac{723 \,\mathrm{I}}{10514}-\frac{241 \sqrt{3}}{10514}\right) \sqrt{751}+\mathrm{I} \sqrt{3}+1\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{3073}\right) \left(-\frac{\mathrm{I} \sqrt{102 \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(-18 \sqrt{751}\, \sqrt{3}+150\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+3468 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}-32946}}{408}+\frac{\left(-50 \,2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}-1156 \left(100+12 \sqrt{751}\, \sqrt{3}\right)^{\frac{1}{3}}-5491\right) \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}}{19455480}+\frac{\sqrt{\left(204272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-7200588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+104829837+\left(28538 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2077266 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{1080860}+\frac{7}{8}\right)^{-n}+\left(\left(\left(2^{\frac{5}{6}} \left(\left(\mathrm{I} \sqrt{3}+\frac{447}{21511}\right) \sqrt{751}-\frac{184075 \,\mathrm{I}}{21511}-\frac{92373 \sqrt{3}}{21511}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}-\frac{850 \,2^{\frac{1}{6}} \left(\left(\mathrm{I} \sqrt{3}-\frac{1419}{25}\right) \sqrt{751}+\frac{243583 \,\mathrm{I}}{25}+\frac{2253 \sqrt{3}}{25}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}-\frac{26144096 \sqrt{2}\, \left(-\frac{423 \sqrt{751}}{11308}+\mathrm{I}\right)}{21511}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}-\frac{13337775 \,2^{\frac{5}{6}} \left(\left(-\frac{91 \,\mathrm{I}}{317}+\frac{1559 \sqrt{3}}{2853}\right) \sqrt{751}+\mathrm{I} \sqrt{3}+\frac{5257}{317}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}-\frac{487817550 \left(\left(\frac{\mathrm{I}}{341}-\frac{313 \sqrt{3}}{3069}\right) \sqrt{751}+\mathrm{I} \sqrt{3}+\frac{2253}{341}\right) 2^{\frac{1}{6}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}+\frac{8690114400 \sqrt{2}\, \sqrt{3}\, \left(\frac{13 \sqrt{751}}{804}+\mathrm{I}\right)}{21511}\right) \sqrt{17 \sqrt{3}\, \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(-3 \sqrt{751}\, \sqrt{3}+25\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+578 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}-5491}+\left(-\frac{727260 \left(\left(\mathrm{I} \sqrt{3}+\frac{645}{713}\right) \sqrt{751}+\frac{50317 \,\mathrm{I}}{713}-\frac{751 \sqrt{3}}{2139}\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}+\frac{5063280 \,2^{\frac{2}{3}} \left(\left(\mathrm{I} \sqrt{3}-\frac{6}{73}\right) \sqrt{751}-\frac{5257 \,\mathrm{I}}{146}+\frac{3755 \sqrt{3}}{438}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21511}+\frac{2358240 \left(\mathrm{I} \sqrt{751}-\frac{8261}{3}\right) \sqrt{3}}{21511}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\frac{15327294180 \left(\left(\frac{4503 \,\mathrm{I}}{80357}+\frac{1501 \sqrt{3}}{80357}\right) \sqrt{751}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21511}+\frac{6623683017600}{21511}+\frac{9740710320 \,2^{\frac{2}{3}} \left(\left(-\frac{723 \,\mathrm{I}}{10514}+\frac{241 \sqrt{3}}{10514}\right) \sqrt{751}+\mathrm{I} \sqrt{3}-1\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{3073}\right) \left(\frac{\mathrm{I} \sqrt{102 \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(-18 \sqrt{751}\, \sqrt{3}+150\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+3468 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}-32946}}{408}+\frac{\left(-50 \,2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}-1156 \left(100+12 \sqrt{751}\, \sqrt{3}\right)^{\frac{1}{3}}-5491\right) \sqrt{\left(816 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-28764 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+418761+\left(114 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+8298 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}}{19455480}+\frac{\sqrt{\left(204272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-7200588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+104829837+\left(28538 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2077266 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{1080860}+\frac{7}{8}\right)^{-n}\right) \left(\left(\left(\frac{646 \,2^{\frac{1}{6}} \left(\sqrt{3}-\frac{487 \sqrt{751}}{14269}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{29}+2^{\frac{5}{6}} \left(\sqrt{3}+\frac{3419 \sqrt{751}}{21779}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}-\frac{182648 \sqrt{751}\, \sqrt{2}}{21779}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\frac{476850 \,2^{\frac{1}{6}} \left(\sqrt{751}\, \sqrt{3}+751\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{21779}+\frac{939675 \left(\sqrt{751}\, \sqrt{3}-\frac{751}{67}\right) 2^{\frac{5}{6}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{21779}-\frac{32425800 \sqrt{2}\, \sqrt{751}\, \sqrt{3}}{21779}\right) \sqrt{17 \sqrt{3}\, \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}+\left(-3 \sqrt{751}\, \sqrt{3}+25\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}+578 \,2^{\frac{2}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}-5491}+\frac{661486320}{29}+\left(\frac{11560 \,2^{\frac{2}{3}} \left(\sqrt{3}+\frac{3 \sqrt{751}}{751}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}}{29}+\frac{680 \left(\sqrt{3}-\frac{201 \sqrt{751}}{751}\right) 2^{\frac{1}{3}} \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}{29}+\frac{393040 \sqrt{3}}{29}\right) \sqrt{\left(272 \sqrt{751}\, 2^{\frac{2}{3}} \sqrt{3}-9588 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{1}{3}}+139587+\left(38 \sqrt{751}\, \sqrt{3}\, 2^{\frac{1}{3}}+2766 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{751}\, \sqrt{3}+25\right)^{\frac{2}{3}}}\right)}{10515541689980926156800}\)
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This specification was found using the strategy pack "Point Placements" and has 76 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \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_{10}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{10}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{10}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{10}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{10}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{10}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{63}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{10}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{10}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{10}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{10}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{64}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{10}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{72}\! \left(x \right)\\ \end{align*}\)