Av(1234, 1324, 2143, 2413, 3142)
Generating Function
\(\displaystyle \frac{2 x -1}{2 x^{4}+x^{3}-x^{2}+3 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 156, 444, 1269, 3629, 10374, 29650, 84743, 242211, 692288, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}+x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)-2 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = 2 a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = 2 a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{2068379 \left(\left(\left(2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-\frac{5139}{101}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+\left(-\frac{427 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}}{3232}-\frac{5139 \,2^{\frac{2}{3}}}{3232}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\frac{496 \sqrt{571}\, \sqrt{3}}{101}\right) \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}-\frac{479640 \,2^{\frac{1}{3}} \left(\sqrt{3}+\frac{9 \sqrt{571}}{571}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{101}+\frac{419685 \left(\sqrt{3}-\frac{369 \sqrt{571}}{3997}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{404}+\frac{282240 \sqrt{571}}{101}\right) \sqrt{48 \sqrt{3}\, \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(-9 \sqrt{571}\, \sqrt{3}+255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-7296}+\frac{1169408 \sqrt{3}\, \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{101}-\frac{8840724480}{101}+\left(\left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}} \left(\frac{70804 \sqrt{3}}{101}-\frac{8292 \sqrt{571}}{101}\right)+\frac{255808 \left(\sqrt{3}-\frac{33 \sqrt{571}}{3997}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{101}\right) \sqrt{\left(19 \sqrt{571}\, \sqrt{3}+827\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\left(128 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}-5248 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+25696 \,2^{\frac{1}{3}}}\right) \left(-\frac{3394838200320 \left(\frac{\sqrt{48 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(9 \sqrt{571}\, \sqrt{3}-255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}-1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+7296}}{192}-\frac{19 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{60480}+\frac{\left(-85 \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}-256 \left(340+12 \sqrt{571}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(57 \sqrt{571}\, \sqrt{3}+2481\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+77088 \,2^{\frac{1}{3}}}}{1935360}+\frac{\sqrt{\left(10849 \sqrt{571}\, \sqrt{3}+472217\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+73088 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+14672416 \,2^{\frac{1}{3}}}\, \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{215040}-\frac{1}{8}\right)^{-n}}{20479}+\left(\left(\left(\left(\frac{5767671 \,2^{\frac{2}{3}}}{655328}+\frac{375471 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}}{655328}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\left(-\frac{96129 \sqrt{571}\, \sqrt{3}\, 2^{\frac{1}{3}}}{20479}+\frac{4726167 \,2^{\frac{1}{3}}}{20479}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-\frac{478416 \sqrt{571}\, \sqrt{3}}{20479}\right) \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}-\frac{383652045 \left(\sqrt{3}-\frac{122387 \sqrt{571}}{1217943}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{81916}+\frac{522327960 \,2^{\frac{1}{3}} \left(\sqrt{3}+\frac{1091 \sqrt{571}}{207273}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}-\frac{271152000 \sqrt{571}}{20479}\right) \sqrt{48 \sqrt{3}\, \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(-9 \sqrt{571}\, \sqrt{3}+255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-7296}-\frac{634988544 \sqrt{3}\, \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{20479}+\left(\left(\frac{703674 \sqrt{571}}{20479}-\frac{3429426 \sqrt{3}}{20479}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}-\frac{28559136 \left(-\frac{9423 \sqrt{571}}{297491}+\sqrt{3}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}\right) \sqrt{\left(19 \sqrt{571}\, \sqrt{3}+827\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\left(128 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}-5248 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+25696 \,2^{\frac{1}{3}}}-\frac{67639320 \left(\sqrt{571}\, \sqrt{3}-\frac{200421}{8947}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{20479}-\frac{3686582108160}{20479}-\frac{75237120 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-\frac{56529}{311}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}\right) \left(-\frac{\sqrt{48 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(9 \sqrt{571}\, \sqrt{3}-255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}-1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+7296}}{192}-\frac{19 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{60480}+\frac{\left(-85 \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}-256 \left(340+12 \sqrt{571}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(57 \sqrt{571}\, \sqrt{3}+2481\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+77088 \,2^{\frac{1}{3}}}}{1935360}+\frac{\sqrt{\left(10849 \sqrt{571}\, \sqrt{3}+472217\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+73088 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+14672416 \,2^{\frac{1}{3}}}\, \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{215040}-\frac{1}{8}\right)^{-n}+\left(\left(\left(-\frac{299171 \left(\left(-\frac{27087 \,\mathrm{I}}{299171}-\frac{371233 \sqrt{3}}{299171}\right) \sqrt{571}+\mathrm{I} \sqrt{3}-\frac{12349017}{299171}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{1310656}-\left(\left(\frac{66994 \sqrt{3}}{20479}+\frac{381 \,\mathrm{I}}{20479}\right) \sqrt{571}+\mathrm{I} \sqrt{3}-\frac{2888118}{20479}\right) 2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-\frac{593000 \sqrt{3}\, \left(\frac{35611 \sqrt{571}}{74125}+\mathrm{I}\right)}{20479}\right) \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}-\frac{2451645 \left(\left(\mathrm{I} \sqrt{3}-\frac{155261}{7783}\right) \sqrt{571}-\frac{216081 \,\mathrm{I}}{7783}+\frac{1257913 \sqrt{3}}{7783}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{163832}-\frac{32760 \left(\left(\mathrm{I} \sqrt{3}+\frac{1832}{13}\right) \sqrt{571}-\frac{21123 \,\mathrm{I}}{13}-\frac{155312 \sqrt{3}}{13}\right) 2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}+\frac{3632247360 \,\mathrm{I}}{20479}-\frac{179726400 \sqrt{571}}{20479}\right) \sqrt{48 \sqrt{3}\, \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(-9 \sqrt{571}\, \sqrt{3}+255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-7296}-\frac{159744 \sqrt{3}\, \left(\mathrm{I} \sqrt{571}+\frac{207273}{26}\right) \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{20479}+\left(\left(\left(\frac{7130547}{20479}-\frac{49299 \,\mathrm{I} \sqrt{3}}{20479}\right) \sqrt{571}-\frac{59596983 \sqrt{3}}{20479}-\frac{950715 \,\mathrm{I}}{20479}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\frac{220080 \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}} \left(\left(\mathrm{I} \sqrt{3}+\frac{45393}{4585}\right) \sqrt{571}-\frac{217551 \,\mathrm{I}}{4585}-\frac{4654221 \sqrt{3}}{4585}\right) 2^{\frac{2}{3}}}{20479}\right) \sqrt{\left(19 \sqrt{571}\, \sqrt{3}+827\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\left(128 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}-5248 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+25696 \,2^{\frac{1}{3}}}+\frac{757591380 \left(\left(-\frac{8947 \,\mathrm{I}}{66807}+\frac{8947 \sqrt{3}}{200421}\right) \sqrt{571}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{20479}-\frac{3686582108160}{20479}-\frac{6837747840 \left(\left(-\frac{311 \,\mathrm{I}}{18843}-\frac{311 \sqrt{3}}{56529}\right) \sqrt{571}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}\right) \left(-\frac{\mathrm{I} \sqrt{48 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(-9 \sqrt{571}\, \sqrt{3}+255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-7296}}{192}+\frac{19 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{60480}+\frac{\left(85 \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+256 \left(340+12 \sqrt{571}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(57 \sqrt{571}\, \sqrt{3}+2481\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+77088 \,2^{\frac{1}{3}}}}{1935360}-\frac{\sqrt{\left(10849 \sqrt{571}\, \sqrt{3}+472217\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+73088 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+14672416 \,2^{\frac{1}{3}}}\, \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{215040}-\frac{1}{8}\right)^{-n}+\left(\frac{\mathrm{I} \sqrt{48 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(-9 \sqrt{571}\, \sqrt{3}+255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-7296}}{192}+\frac{19 \sqrt{768 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+154176+\left(57 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+2481 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{60480}+\frac{\left(85 \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+256 \left(340+12 \sqrt{571}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(57 \sqrt{571}\, \sqrt{3}+2481\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+77088 \,2^{\frac{1}{3}}}}{1935360}-\frac{\sqrt{\left(10849 \sqrt{571}\, \sqrt{3}+472217\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+73088 \,2^{\frac{2}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+14672416 \,2^{\frac{1}{3}}}\, \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{215040}-\frac{1}{8}\right)^{-n} \left(\left(\left(\frac{299171 \left(\left(-\frac{27087 \,\mathrm{I}}{299171}+\frac{371233 \sqrt{3}}{299171}\right) \sqrt{571}+\mathrm{I} \sqrt{3}+\frac{12349017}{299171}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{1310656}+2^{\frac{1}{3}} \left(\left(\frac{381 \,\mathrm{I}}{20479}-\frac{66994 \sqrt{3}}{20479}\right) \sqrt{571}+\mathrm{I} \sqrt{3}+\frac{2888118}{20479}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+\frac{593000 \sqrt{3}\, \left(-\frac{35611 \sqrt{571}}{74125}+\mathrm{I}\right)}{20479}\right) \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\frac{2451645 \left(\left(\mathrm{I} \sqrt{3}+\frac{155261}{7783}\right) \sqrt{571}-\frac{216081 \,\mathrm{I}}{7783}-\frac{1257913 \sqrt{3}}{7783}\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{163832}+\frac{32760 \,2^{\frac{1}{3}} \left(\left(\mathrm{I} \sqrt{3}-\frac{1832}{13}\right) \sqrt{571}-\frac{21123 \,\mathrm{I}}{13}+\frac{155312 \sqrt{3}}{13}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}-\frac{3632247360 \,\mathrm{I}}{20479}-\frac{179726400 \sqrt{571}}{20479}\right) \sqrt{48 \sqrt{3}\, \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}+\left(-9 \sqrt{571}\, \sqrt{3}+255\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+1536 \,2^{\frac{1}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}-7296}+\frac{159744 \sqrt{3}\, \left(\mathrm{I} \sqrt{571}-\frac{207273}{26}\right) \sqrt{256 \,2^{\frac{1}{3}} \left(\sqrt{571}\, \sqrt{3}-41\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+51392+\left(19 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}+827 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}}{20479}+\left(\left(\left(\frac{7130547}{20479}+\frac{49299 \,\mathrm{I} \sqrt{3}}{20479}\right) \sqrt{571}-\frac{59596983 \sqrt{3}}{20479}+\frac{950715 \,\mathrm{I}}{20479}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}-\frac{220080 \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}} \left(\left(\mathrm{I} \sqrt{3}-\frac{45393}{4585}\right) \sqrt{571}-\frac{217551 \,\mathrm{I}}{4585}+\frac{4654221 \sqrt{3}}{4585}\right) 2^{\frac{2}{3}}}{20479}\right) \sqrt{\left(19 \sqrt{571}\, \sqrt{3}+827\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}+\left(128 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{571}-5248 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}+25696 \,2^{\frac{1}{3}}}-\frac{757591380 \left(\left(-\frac{8947 \,\mathrm{I}}{66807}-\frac{8947 \sqrt{3}}{200421}\right) \sqrt{571}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{2}{3}}}{20479}-\frac{3686582108160}{20479}+\frac{6837747840 \,2^{\frac{1}{3}} \left(\left(-\frac{311 \,\mathrm{I}}{18843}+\frac{311 \sqrt{3}}{56529}\right) \sqrt{571}+\mathrm{I} \sqrt{3}-1\right) \left(3 \sqrt{571}\, \sqrt{3}+85\right)^{\frac{1}{3}}}{20479}\right)\right)}{120051316732832671334400}\)
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_{16}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{51}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{71}\! \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_{62}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{70}\! \left(x \right)\\
\end{align*}\)