Av(1234, 1342, 1432, 2314, 4213)
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Generating Function
\(\displaystyle \frac{3 x^{6}+11 x^{5}+6 x^{4}+x^{3}-x +1}{\left(x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 49, 105, 225, 487, 1041, 2209, 4695, 9981, 21197, 45007, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+3 x^{6}+11 x^{5}+6 x^{4}+x^{3}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 105\)
\(\displaystyle a \! \left(n +4\right) = 2 a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)+21, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-\sqrt{2}\, \left(172235 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+5954454 \sqrt{53}\right) \sqrt{3}+251370 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{53}-18591552\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(611363268-841 \sqrt{2}\, \left(4655 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+15894 \sqrt{53}\right) \sqrt{3}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+5299141 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}+698013180\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}-76821 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(37 \sqrt{3}\, \sqrt{2}-54 \sqrt{53}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-64606461 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}\, \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-834615969 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}-36436287996\right) \left(-\frac{\mathrm{I} \sqrt{522 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+\left(-54 \sqrt{53}\, \sqrt{3}\, \sqrt{2}+222\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+5046 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+5046}}{348}+\frac{\left(1682 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+74 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-841\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}}{878004}-\frac{\sqrt{\left(-92220+36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-636 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+121476 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+3476694}\, \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}}{48778}-\frac{1}{4}\right)^{-n}}{485817173280}\\+\\\frac{\left(\left(\left(\sqrt{2}\, \left(172235 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-5954454 \sqrt{53}\right) \sqrt{3}-251370 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{53}-18591552\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(611363268+841 \sqrt{2}\, \left(4655 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-15894 \sqrt{53}\right) \sqrt{3}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-5299141 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}+698013180\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+76821 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(37 \sqrt{3}\, \sqrt{2}-54 \sqrt{53}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+64606461 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}\, \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+834615969 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}-36436287996\right) \left(\frac{\mathrm{I} \sqrt{522 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+\left(-54 \sqrt{53}\, \sqrt{3}\, \sqrt{2}+222\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+5046 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+5046}}{348}+\frac{\left(1682 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+74 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-841\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}}{878004}-\frac{\sqrt{\left(-92220+36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-636 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+121476 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+3476694}\, \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}}{48778}-\frac{1}{4}\right)^{-n}}{485817173280}\\+\\\frac{\left(\left(\left(2 \sqrt{53}\, \left(25646 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+2977227 \sqrt{2}\right) \sqrt{3}-10494 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}+18591552\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(-58 \sqrt{53}\, \left(1016 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-230463 \sqrt{2}\right) \sqrt{3}+2517606 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}-611363268\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-3917378 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}-698013180\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+27666 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(-56 \sqrt{53}\, \sqrt{3}+1143 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-802314 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(22 \sqrt{53}\, \sqrt{3}+45 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+4829022 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}-36436287996\right) \left(-\frac{\sqrt{522 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+\left(54 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-222\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-5046 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-5046}}{348}+\frac{\left(-1682 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-74 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}}{878004}+\frac{\sqrt{\left(-92220+36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-636 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+121476 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+3476694}\, \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}}{48778}-\frac{1}{4}\right)^{-n}}{485817173280}\\-\frac{21}{5}+\\\frac{\left(\left(\left(-2 \sqrt{53}\, \left(25646 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-2977227 \sqrt{2}\right) \sqrt{3}+10494 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}+18591552\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(58 \sqrt{53}\, \left(1016 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+230463 \sqrt{2}\right) \sqrt{3}-2517606 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}-611363268\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+3917378 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}-698013180\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}-27666 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(-56 \sqrt{53}\, \sqrt{3}+1143 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+802314 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(22 \sqrt{53}\, \sqrt{3}+45 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-4829022 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+87 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}-36436287996\right) \left(\frac{\sqrt{522 \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}+\left(54 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-222\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-5046 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-5046}}{348}+\frac{\left(-1682 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-74 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841\right) \sqrt{\left(-290+116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-2 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+382 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+10933}}{878004}+\frac{\sqrt{\left(-92220+36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-636 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+121476 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+3476694}\, \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}}{48778}-\frac{1}{4}\right)^{-n}}{485817173280} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 101 rules.

Found on January 18, 2022.

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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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \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_{10}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \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_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{21}\! \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_{21}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 0\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \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_{4}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= 2 F_{43}\! \left(x \right)+F_{62}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{65}\! \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_{37}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{84}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= x^{2}\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= 2 F_{43}\! \left(x \right)+F_{93}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{69}\! \left(x \right)\\ \end{align*}\)