Av(1243, 1342, 1432, 3142, 3214)
Generating Function
\(\displaystyle \frac{\left(x -1\right)^{3}}{x^{4}+4 x^{3}-5 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 151, 411, 1128, 3116, 8619, 23819, 65773, 181589, 501386, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+4 a \! \left(n +1\right)-5 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+4 a \! \left(n +1\right)-5 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle -\frac{89 \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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}\right) \left(\left(\left(\left(\frac{2381 \left(\left(\mathrm{I} \sqrt{17}-\frac{15351}{2381}\right) \sqrt{3}-\frac{731 \,\mathrm{I}}{2381}+\frac{5943 \sqrt{17}}{2381}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\left(\left(\frac{1785}{89}+\mathrm{I} \sqrt{17}\right) \sqrt{3}+\frac{861 \sqrt{17}}{89}-\frac{2941 \,\mathrm{I}}{89}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-\frac{15120 \sqrt{17}}{89}+\frac{97580 \,\mathrm{I}}{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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{53856 \left(\left(\mathrm{I}-\frac{223 \sqrt{17}}{816}\right) \sqrt{3}-\frac{128 \,\mathrm{I} \sqrt{17}}{51}+\frac{209}{48}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{1884960 \left(\left(\mathrm{I}-\frac{77 \sqrt{17}}{816}\right) \sqrt{3}-\frac{2 \,\mathrm{I} \sqrt{17}}{51}-\frac{7}{48}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{26114550 \left(-\frac{45 \sqrt{17}}{323}+\mathrm{I}\right) \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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}}{89}+\frac{551905}{89}\right) \sqrt{3}+\frac{9435 \,\mathrm{I}}{89}-\frac{229590 \sqrt{17}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-\frac{1960 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-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(\mathrm{I}-\frac{237 \sqrt{17}}{2873}\right) \sqrt{3}+\frac{711 \,\mathrm{I} \sqrt{17}}{2873}-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(\mathrm{I}-\frac{81 \sqrt{17}}{425}\right) \sqrt{3}-\frac{243 \,\mathrm{I} \sqrt{17}}{425}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}+\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}+\left(\left(\left(\frac{2381 \,5^{\frac{2}{3}} \left(\left(\mathrm{I} \sqrt{17}+\frac{15351}{2381}\right) \sqrt{3}-\frac{731 \,\mathrm{I}}{2381}-\frac{5943 \sqrt{17}}{2381}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\left(\left(-\frac{1785}{89}+\mathrm{I} \sqrt{17}\right) \sqrt{3}-\frac{861 \sqrt{17}}{89}-\frac{2941 \,\mathrm{I}}{89}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+\frac{15120 \sqrt{17}}{89}+\frac{97580 \,\mathrm{I}}{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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{53856 \left(\left(\mathrm{I}+\frac{223 \sqrt{17}}{816}\right) \sqrt{3}-\frac{128 \,\mathrm{I} \sqrt{17}}{51}-\frac{209}{48}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{1884960 \left(\left(\mathrm{I}+\frac{77 \sqrt{17}}{816}\right) \sqrt{3}-\frac{2 \,\mathrm{I} \sqrt{17}}{51}+\frac{7}{48}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{26114550 \sqrt{3}\, \left(\frac{45 \sqrt{17}}{323}+\mathrm{I}\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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}}{89}+\frac{551905}{89}\right) \sqrt{3}-\frac{9435 \,\mathrm{I}}{89}-\frac{229590 \sqrt{17}}{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) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-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(\mathrm{I}+\frac{237 \sqrt{17}}{2873}\right) \sqrt{3}+\frac{711 \,\mathrm{I} \sqrt{17}}{2873}+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(\mathrm{I}+\frac{81 \sqrt{17}}{425}\right) \sqrt{3}-\frac{243 \,\mathrm{I} \sqrt{17}}{425}-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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}+\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}+\left(\left(\left(\frac{11558 \,\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{17}\, \sqrt{3}-\frac{14229}{5779}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\frac{171360 \,\mathrm{I}}{89}-2 \,\mathrm{I} \left(\sqrt{17}\, \sqrt{3}+\frac{6579}{89}\right) 5^{\frac{1}{3}} \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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{303688 \,\mathrm{I} \left(-\frac{654 \sqrt{17}}{493}+\sqrt{3}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{5393080 \,\mathrm{I} \left(\sqrt{3}+\frac{48 \sqrt{17}}{1751}\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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(-\frac{85510 \sqrt{3}}{89}-\frac{71220 \sqrt{17}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{234430 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{3}} \left(\sqrt{3}-\frac{948 \sqrt{17}}{3349}\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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-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 \,5^{\frac{1}{3}} \left(\sqrt{17}\, \sqrt{3}-\frac{425}{81}\right) \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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}-\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+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{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}-\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}}{89}\right)}{4373768415988125000}\)
This specification was found using the strategy pack "Point Placements" and has 72 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 72 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_{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_{14}\! \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_{43}\! \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_{26}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{31}\! \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_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \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_{40}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
\end{align*}\)