Av(1342, 1432, 2413, 3124, 3142)
View Raw Data
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{2}+2 x -1\right)}{x^{4}-3 x^{2}+4 x -1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 59, 181, 553, 1688, 5152, 15725, 47997, 146501, 447165, 1364882, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-3 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}+2 x -1\right) = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = 3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 4\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle -\frac{247 \left(\left(\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-\frac{3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}-\frac{27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}}{2}-\frac{54}{19}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-\frac{78 \sqrt{61}\, \sqrt{3}}{19}+\frac{1053}{19}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{117 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{19}+\frac{135}{19}\right) \sqrt{-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+36-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}-\frac{9882}{19}+\left(\left(\frac{15 \sqrt{61}\, \sqrt{3}}{38}-\frac{183}{38}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{39 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{61}\, \sqrt{3}}{38}-\frac{549 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{38}+\frac{549}{19}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}\right) \left(\left(\left(\left(\left(\left(\frac{263 \,\mathrm{I}}{793}+\frac{229 \sqrt{3}}{793}\right) \sqrt{61}-\frac{51}{13}-\frac{19 \,\mathrm{I} \sqrt{3}}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{3 \sqrt{3}}{61}-\frac{147 \,\mathrm{I}}{793}\right) \sqrt{61}+\frac{3}{13}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{6 \,\mathrm{I} \sqrt{61}}{61}-\frac{54}{13}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(-\frac{120 \sqrt{3}}{793}+\frac{252 \,\mathrm{I}}{793}\right) \sqrt{61}-\frac{12 \,\mathrm{I} \sqrt{3}}{13}+\frac{24}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{2412 \,\mathrm{I}}{793}-\frac{24 \sqrt{3}}{61}\right) \sqrt{61}+\frac{180 \,\mathrm{I} \sqrt{3}}{13}+\frac{72}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-36+\frac{324 \,\mathrm{I} \sqrt{61}}{793}\right) \sqrt{-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+36-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\left(-\frac{918 \sqrt{3}}{793}-\frac{750 \,\mathrm{I}}{793}\right) \sqrt{61}+\frac{51 \,\mathrm{I} \sqrt{3}}{13}+\frac{207}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{36 \sqrt{3}}{61}+\frac{1584 \,\mathrm{I}}{793}\right) \sqrt{61}-\frac{123 \,\mathrm{I} \sqrt{3}}{13}-\frac{81}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{594}{13}-\frac{36 \,\mathrm{I} \sqrt{61}}{61}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(\frac{2268 \sqrt{3}}{793}-\frac{6804 \,\mathrm{I}}{793}\right) \sqrt{61}+36 \,\mathrm{I} \sqrt{3}-36\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{4752}{13}+\left(\left(\frac{4140 \sqrt{3}}{793}+\frac{12420 \,\mathrm{I}}{793}\right) \sqrt{61}-\frac{972}{13}-\frac{972 \,\mathrm{I} \sqrt{3}}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}-\frac{\mathrm{I} \sqrt{27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-36+3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}-\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}+\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}\right)^{-n}+\left(\left(\left(\left(\left(-\frac{263 \,\mathrm{I}}{793}+\frac{229 \sqrt{3}}{793}\right) \sqrt{61}-\frac{51}{13}+\frac{19 \,\mathrm{I} \sqrt{3}}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{3 \sqrt{3}}{61}+\frac{147 \,\mathrm{I}}{793}\right) \sqrt{61}+\frac{3}{13}-\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{6 \,\mathrm{I} \sqrt{61}}{61}-\frac{54}{13}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(-\frac{120 \sqrt{3}}{793}-\frac{252 \,\mathrm{I}}{793}\right) \sqrt{61}+\frac{12 \,\mathrm{I} \sqrt{3}}{13}+\frac{24}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{2412 \,\mathrm{I}}{793}-\frac{24 \sqrt{3}}{61}\right) \sqrt{61}-\frac{180 \,\mathrm{I} \sqrt{3}}{13}+\frac{72}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-36-\frac{324 \,\mathrm{I} \sqrt{61}}{793}\right) \sqrt{-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+36-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\left(-\frac{918 \sqrt{3}}{793}+\frac{750 \,\mathrm{I}}{793}\right) \sqrt{61}-\frac{51 \,\mathrm{I} \sqrt{3}}{13}+\frac{207}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{36 \sqrt{3}}{61}-\frac{1584 \,\mathrm{I}}{793}\right) \sqrt{61}+\frac{123 \,\mathrm{I} \sqrt{3}}{13}-\frac{81}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{594}{13}+\frac{36 \,\mathrm{I} \sqrt{61}}{61}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(\frac{2268 \sqrt{3}}{793}+\frac{6804 \,\mathrm{I}}{793}\right) \sqrt{61}-36 \,\mathrm{I} \sqrt{3}-36\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{4752}{13}+\left(\left(\frac{4140 \sqrt{3}}{793}-\frac{12420 \,\mathrm{I}}{793}\right) \sqrt{61}-\frac{972}{13}+\frac{972 \,\mathrm{I} \sqrt{3}}{13}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\mathrm{I} \sqrt{27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-36+3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}-\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}+\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}\right)^{-n}+\left(\left(\left(\left(-\frac{114}{13}+\frac{518 \sqrt{61}\, \sqrt{3}}{793}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{6 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{61}\, \sqrt{3}}{61}-\frac{36}{13}-\frac{30 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-\frac{48}{13}+\frac{240 \sqrt{61}\, \sqrt{3}}{793}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{144 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}+\frac{48 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{61}\, \sqrt{3}}{61}-\frac{360}{13}\right) \sqrt{-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+36-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\frac{558}{13}-\frac{2556 \sqrt{61}\, \sqrt{3}}{793}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{270 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}-\frac{72 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{61}\, \sqrt{3}}{61}+\frac{324}{13}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-\frac{4536 \sqrt{61}\, \sqrt{3}}{793}+72\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{4752}{13}-\frac{8280 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{61}\, \sqrt{3}}{793}+\frac{1944 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}\right) \left(\frac{\left(-\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}-\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}-\frac{\sqrt{-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+36-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}\right)^{-n}+\frac{1296 \left(\frac{\left(-\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}-\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}+\frac{\sqrt{-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+36-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}\right)^{-n}}{13}\right)}{51228288}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 33 rules.

Found on January 18, 2022.

Finding the specification took 0 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 33 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_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{7}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \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_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)\\ \end{align*}\)