Av(1234, 1342, 1432, 2314, 3241)
Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(2 x^{5}-8 x^{4}+3 x^{3}-4 x^{2}+3 x -1\right)}{\left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 114, 253, 554, 1194, 2551, 5438, 11574, 24601, 52262, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-\left(x +1\right) \left(2 x^{5}-8 x^{4}+3 x^{3}-4 x^{2}+3 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 114\)
\(\displaystyle a \! \left(n +4\right) = 2 a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)+10 n +9, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 114\)
\(\displaystyle a \! \left(n +4\right) = 2 a \! \left(n \right)+2 a \! \left(n +1\right)+a \! \left(n +2\right)+a \! \left(n +3\right)+10 n +9, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-\sqrt{2}\, \left(12395 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+440046 \sqrt{53}\right) \sqrt{3}+18090 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{53}-2434608\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(46534212-841 \sqrt{2}\, \left(335 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+1566 \sqrt{53}\right) \sqrt{3}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+470119 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}+116335530\right) \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}-7569 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(37 \sqrt{2}\, \sqrt{3}-54 \sqrt{53}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-6365529 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+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}}-61679781 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}+6072714666\right) \left(-\frac{\mathrm{I} \sqrt{522 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-92220\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}}{40484764440}\\+\\\frac{\left(\left(\left(\sqrt{2}\, \left(12395 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-440046 \sqrt{53}\right) \sqrt{3}-18090 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{53}-2434608\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(46534212+841 \sqrt{2}\, \left(335 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-1566 \sqrt{53}\right) \sqrt{3}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-470119 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}+116335530\right) \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+6365529 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+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}}+7569 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(37 \sqrt{2}\, \sqrt{3}-54 \sqrt{53}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+61679781 \,\mathrm{I} \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}\, \sqrt{3}+6072714666\right) \left(\frac{\mathrm{I} \sqrt{522 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-92220\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}}{40484764440}\\+\\\frac{\left(\left(\left(\sqrt{53}\, \left(3743 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+440046 \sqrt{2}\right) \sqrt{3}-3816 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}+2434608\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(-29 \sqrt{53}\, \left(83 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-45414 \sqrt{2}\right) \sqrt{3}+179829 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}-46534212\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-283417 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}-116335530\right) \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+2523 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(-73 \sqrt{53}\, \sqrt{3}+954 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}-2523 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(499 \sqrt{53}\, \sqrt{3}+2385 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+1390173 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}+6072714666\right) \left(-\frac{\sqrt{522 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-92220\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}}{40484764440}\\+\\\frac{\left(\left(\left(-\sqrt{53}\, \left(3743 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}-440046 \sqrt{2}\right) \sqrt{3}+3816 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}+2434608\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+\left(29 \sqrt{53}\, \left(83 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}+45414 \sqrt{2}\right) \sqrt{3}-179829 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{2}-46534212\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+283417 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}-116335530\right) \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}-2523 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(-73 \sqrt{53}\, \sqrt{3}+954 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+2523 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \left(499 \sqrt{53}\, \sqrt{3}+2385 \sqrt{2}\right) \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-1390173 \sqrt{-9 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}} \sqrt{53}\, \sqrt{3}\, \sqrt{2}+87 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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}+37 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{2}{3}}+841 \left(37+9 \sqrt{53}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+841}\, \sqrt{3}\, \sqrt{53}+6072714666\right) \left(\frac{\sqrt{522 \sqrt{\left(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(116 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-290\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(36888 \sqrt{53}\, \sqrt{3}\, \sqrt{2}-92220\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}}{40484764440}\\-2 n -\frac{3}{5} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 186 rules.
Found on January 18, 2022.Finding the specification took 6 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{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_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \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_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{17}\! \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_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{92}\! \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_{76}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \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_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \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_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{185}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{125}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= x^{2}\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{176}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{161}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{161}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{151}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{151}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)