Av(1234, 1324, 1342, 3142, 3214)
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Generating Function
\(\displaystyle \frac{\left(x -1\right)^{3}}{x^{4}+4 x^{3}-5 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 151, 411, 1128, 3116, 8619, 23819, 65773, 181589, 501386, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+4 a \! \left(n +1\right)-5 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle -\frac{89 \left(\left(\left(\left(\frac{11558 \,\mathrm{I} \left(\sqrt{17}\, \sqrt{3}-\frac{14229}{5779}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\frac{171360 \,\mathrm{I}}{89}-2 \,\mathrm{I} \,5^{\frac{1}{3}} \left(\sqrt{17}\, \sqrt{3}+\frac{6579}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{303688 \,\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{3}-\frac{654 \sqrt{17}}{493}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{5393080 \,\mathrm{I} \left(\frac{48 \sqrt{17}}{1751}+\sqrt{3}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{43066100 \,\mathrm{I} \sqrt{3}}{89}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}-\frac{583100 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(-\frac{71220 \sqrt{17}}{89}-\frac{85510 \sqrt{3}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{234430 \left(-\frac{948 \sqrt{17}}{3349}+\sqrt{3}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}-\frac{1094940 \left(\sqrt{17}\, \sqrt{3}+\frac{2873}{237}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}+\frac{15201417000}{89}-\frac{13097700 \left(\sqrt{17}\, \sqrt{3}-\frac{425}{81}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \left(-\frac{\mathrm{I} \sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-90 \sqrt{17}\, \sqrt{3}+165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-161700}}{210}+\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}-\frac{\left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}} \sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}}{37730}-1\right)^{-n}+\left(\left(\left(\frac{2381 \left(\left(\mathrm{I} \sqrt{3}+\frac{5943}{2381}\right) \sqrt{17}-\frac{731 \,\mathrm{I}}{2381}-\frac{15351 \sqrt{3}}{2381}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\left(\left(\mathrm{I} \sqrt{3}+\frac{861}{89}\right) \sqrt{17}+\frac{1785 \sqrt{3}}{89}-\frac{2941 \,\mathrm{I}}{89}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+\frac{97580 \,\mathrm{I}}{89}-\frac{15120 \sqrt{17}}{89}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{53856 \,5^{\frac{2}{3}} \left(\left(-\frac{128 \,\mathrm{I}}{51}-\frac{223 \sqrt{3}}{816}\right) \sqrt{17}+\mathrm{I} \sqrt{3}+\frac{209}{48}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{1884960 \,5^{\frac{1}{3}} \left(\left(-\frac{2 \,\mathrm{I}}{51}-\frac{77 \sqrt{3}}{816}\right) \sqrt{17}+\mathrm{I} \sqrt{3}-\frac{7}{48}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{26114550 \sqrt{3}\, \left(\mathrm{I}-\frac{45 \sqrt{17}}{323}\right)}{89}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}-\frac{78400 \sqrt{3}\, \left(\mathrm{I} \sqrt{17}+\frac{119}{32}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(\left(\frac{1090 \,\mathrm{I} \sqrt{3}}{89}-\frac{229590}{89}\right) \sqrt{17}+\frac{551905 \sqrt{3}}{89}+\frac{9435 \,\mathrm{I}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-\frac{1960 \,5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} \left(\mathrm{I} \sqrt{3}\, \sqrt{17}-\frac{51 \,\mathrm{I}}{8}-\frac{2023 \sqrt{3}}{8}-108 \sqrt{17}\right)}{89}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}-\frac{6636630 \,5^{\frac{2}{3}} \left(\left(\frac{711 \,\mathrm{I}}{2873}-\frac{237 \sqrt{3}}{2873}\right) \sqrt{17}+\mathrm{I} \sqrt{3}-1\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}+\frac{15201417000}{89}-\frac{34361250 \left(\left(-\frac{243 \,\mathrm{I}}{425}-\frac{81 \sqrt{3}}{425}\right) \sqrt{17}+\mathrm{I} \sqrt{3}+1\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \left(-\frac{\sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(90 \sqrt{17}\, \sqrt{3}-165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+161700}}{210}-\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(-11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}+\frac{\left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}} \sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}}{37730}-1\right)^{-n}+\left(\left(\left(\frac{2381 \,5^{\frac{2}{3}} \left(\left(\mathrm{I} \sqrt{3}-\frac{5943}{2381}\right) \sqrt{17}-\frac{731 \,\mathrm{I}}{2381}+\frac{15351 \sqrt{3}}{2381}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+5^{\frac{1}{3}} \left(\left(\mathrm{I} \sqrt{3}-\frac{861}{89}\right) \sqrt{17}-\frac{1785 \sqrt{3}}{89}-\frac{2941 \,\mathrm{I}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+\frac{97580 \,\mathrm{I}}{89}+\frac{15120 \sqrt{17}}{89}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{53856 \left(\left(-\frac{128 \,\mathrm{I}}{51}+\frac{223 \sqrt{3}}{816}\right) \sqrt{17}+\mathrm{I} \sqrt{3}-\frac{209}{48}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{1884960 \,5^{\frac{1}{3}} \left(\left(-\frac{2 \,\mathrm{I}}{51}+\frac{77 \sqrt{3}}{816}\right) \sqrt{17}+\mathrm{I} \sqrt{3}+\frac{7}{48}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{26114550 \sqrt{3}\, \left(\mathrm{I}+\frac{45 \sqrt{17}}{323}\right)}{89}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}+\frac{78400 \left(\mathrm{I} \sqrt{17}-\frac{119}{32}\right) \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(\left(-\frac{1090 \,\mathrm{I} \sqrt{3}}{89}-\frac{229590}{89}\right) \sqrt{17}+\frac{551905 \sqrt{3}}{89}-\frac{9435 \,\mathrm{I}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{1960 \left(\mathrm{I} \sqrt{3}\, \sqrt{17}-\frac{51 \,\mathrm{I}}{8}+\frac{2023 \sqrt{3}}{8}+108 \sqrt{17}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}+\frac{6636630 \left(\left(\frac{711 \,\mathrm{I}}{2873}+\frac{237 \sqrt{3}}{2873}\right) \sqrt{17}+\mathrm{I} \sqrt{3}+1\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}+\frac{15201417000}{89}+\frac{34361250 \left(\left(-\frac{243 \,\mathrm{I}}{425}+\frac{81 \sqrt{3}}{425}\right) \sqrt{17}+\mathrm{I} \sqrt{3}-1\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \left(\frac{\sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(90 \sqrt{17}\, \sqrt{3}-165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+161700}}{210}-\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(-11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}+\frac{\left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}} \sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}}{37730}-1\right)^{-n}+\frac{15586263000 \left(\frac{\mathrm{I} \sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-90 \sqrt{17}\, \sqrt{3}+165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-161700}}{210}+\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}-\frac{\left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}} \sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}}{37730}-1\right)^{-n}}{89}\right) \left(\left(\left(\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{17}\, \sqrt{3}-\frac{11}{6}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-\frac{245 \,\mathrm{I} \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{6}+\frac{2695 \,\mathrm{I}}{8}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{46585 \,\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{3}-\frac{18 \sqrt{17}}{11}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{72}-\frac{1037575 \,\mathrm{I} \sqrt{3}\, \left(5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-\frac{61}{11}\right)}{72}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}-\frac{280616875}{4}+\left(\left(\frac{32725 \sqrt{3}}{24}-\frac{10675 \sqrt{17}}{24}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{20825 \left(\sqrt{3}+\frac{11 \sqrt{17}}{17}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{24}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{3}\, 5^{\frac{2}{3}} \sqrt{17}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}\right)}{4373768415988125000}\)
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This specification was found using the strategy pack "Point Placements" and has 46 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{14}\! \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_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \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_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)\\ \end{align*}\)