Av(1234, 1324, 1342, 1432, 2143)
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Generating Function
\(\displaystyle \frac{x^{4}-x^{3}-x^{2}-2 x +1}{x^{4}-x^{3}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 178, 547, 1680, 5160, 15849, 48680, 149520, 459249, 1410578, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-x^{3}-3 x +1\right) F \! \left(x \right)-x^{4}+x^{3}+x^{2}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)-3 a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(\left(-2115 \sqrt{2}\, 3^{\frac{1}{6}}+94 \,3^{\frac{2}{3}} \sqrt{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-47 \sqrt{2}\, 3^{\frac{5}{6}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-1104 \sqrt{3}\, \sqrt{2}\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}-4275 \sqrt{2}\, 3^{\frac{5}{6}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\left(-192375 \sqrt{2}\, 3^{\frac{1}{6}}+8550 \,3^{\frac{2}{3}} \sqrt{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+35100 \sqrt{3}\, \sqrt{2}\right) \sqrt{\sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(2 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-30 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-2 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+3}-1872 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}} \left(\left(\sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-\frac{2359 \,3^{\frac{2}{3}}}{156}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\frac{19 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}}{156}-\frac{337 \,3^{\frac{1}{3}}}{52}\right)\right) \left(\frac{\sqrt{6 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-180 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-12 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+18}}{12}+\frac{\left(60 \,3^{\frac{2}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+4 \left(135+3 \sqrt{337}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+3\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}}{900}-\frac{\sqrt{\left(-10784 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+477192 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-26286+\left(8088 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-110536 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}} 3^{\frac{1}{6}}}{225}+\frac{1}{4}\right)^{-n}}{3639600}\\+\\\frac{\left(\left(\left(\left(-337 \,\mathrm{I} \sqrt{2}\, 3^{\frac{2}{3}}+53 \,\mathrm{I} \sqrt{337}\, 3^{\frac{1}{6}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+\left(-2696 \,\mathrm{I} \sqrt{2}\, 3^{\frac{1}{3}}+121 \,\mathrm{I} \sqrt{337}\, 3^{\frac{5}{6}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-41 \,\mathrm{I} \sqrt{337}\, \sqrt{3}\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+225 \,\mathrm{I} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}} 3^{\frac{5}{6}} \sqrt{337}+\left(-75825 \,\mathrm{I} \sqrt{2}\, 3^{\frac{2}{3}}+10125 \,\mathrm{I} \sqrt{337}\, 3^{\frac{1}{6}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+1575 \,\mathrm{I} \sqrt{3}\, \sqrt{337}\right) \sqrt{\sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(2 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-30 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-2 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+3}+1872 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}} \left(\left(\sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-\frac{2359 \,3^{\frac{2}{3}}}{156}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\frac{19 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}}{156}-\frac{337 \,3^{\frac{1}{3}}}{52}\right)\right) \left(\frac{\mathrm{I} \sqrt{6 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(-12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}+180 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+12 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-18}}{12}+\frac{\left(-60 \,3^{\frac{2}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-4 \left(135+3 \sqrt{337}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-3\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}}{900}+\frac{\sqrt{\left(-10784 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+477192 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-26286+\left(8088 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-110536 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}} 3^{\frac{1}{6}}}{225}+\frac{1}{4}\right)^{-n}}{3639600}\\+\\\frac{\left(\left(\left(\left(2115 \sqrt{2}\, 3^{\frac{1}{6}}-94 \,3^{\frac{2}{3}} \sqrt{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+47 \sqrt{2}\, 3^{\frac{5}{6}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+1104 \sqrt{3}\, \sqrt{2}\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+4275 \sqrt{2}\, 3^{\frac{5}{6}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\left(192375 \sqrt{2}\, 3^{\frac{1}{6}}-8550 \,3^{\frac{2}{3}} \sqrt{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-35100 \sqrt{3}\, \sqrt{2}\right) \sqrt{\sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(2 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-30 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-2 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+3}-1872 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}} \left(\left(\sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-\frac{2359 \,3^{\frac{2}{3}}}{156}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\frac{19 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}}{156}-\frac{337 \,3^{\frac{1}{3}}}{52}\right)\right) \left(-\frac{\sqrt{6 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-180 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-12 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+18}}{12}+\frac{\left(60 \,3^{\frac{2}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+4 \left(135+3 \sqrt{337}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}+3\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}}{900}-\frac{\sqrt{\left(-10784 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+477192 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-26286+\left(8088 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-110536 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}} 3^{\frac{1}{6}}}{225}+\frac{1}{4}\right)^{-n}}{3639600}\\+\\\frac{\left(\left(\left(\left(\mathrm{I} \sqrt{2}\, 3^{\frac{2}{3}}-\frac{53 \,\mathrm{I} \,3^{\frac{1}{6}} \sqrt{337}}{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+\left(8 \,\mathrm{I} \sqrt{2}\, 3^{\frac{1}{3}}-\frac{121 \,\mathrm{I} \,3^{\frac{5}{6}} \sqrt{337}}{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\frac{41 \,\mathrm{I} \sqrt{3}\, \sqrt{337}}{337}\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}-\frac{225 \,\mathrm{I} \,3^{\frac{5}{6}} \sqrt{337}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}}{337}+\left(225 \,\mathrm{I} \sqrt{2}\, 3^{\frac{2}{3}}-\frac{10125 \,\mathrm{I} \,3^{\frac{1}{6}} \sqrt{337}}{337}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-\frac{1575 \,\mathrm{I} \sqrt{3}\, \sqrt{337}}{337}\right) \sqrt{\sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(2 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-30 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-2 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+3}+\frac{1872 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}} \left(\left(\sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-\frac{2359 \,3^{\frac{2}{3}}}{156}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}+\frac{19 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}}{156}-\frac{337 \,3^{\frac{1}{3}}}{52}\right)}{337}\right) \left(-\frac{\mathrm{I} \sqrt{6 \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}+\left(-12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}+180 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}+12 \,3^{\frac{1}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-18}}{12}+\frac{\left(-60 \,3^{\frac{2}{3}} \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}-4 \left(135+3 \sqrt{337}\, \sqrt{3}\, \sqrt{2}\right)^{\frac{1}{3}}-3\right) \sqrt{\left(-16 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+708 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-39+\left(12 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-164 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}}{900}+\frac{\sqrt{\left(-10784 \sqrt{2}\, \sqrt{337}\, 3^{\frac{5}{6}}+477192 \,3^{\frac{1}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{1}{3}}-26286+\left(8088 \sqrt{2}\, 3^{\frac{1}{6}} \sqrt{337}-110536 \,3^{\frac{2}{3}}\right) \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}}}\, \left(\sqrt{337}\, \sqrt{3}\, \sqrt{2}+45\right)^{\frac{2}{3}} 3^{\frac{1}{6}}}{225}+\frac{1}{4}\right)^{-n}}{10800} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 73 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{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_{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_{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_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{67}\! \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_{31}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= x^{2}\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{52}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{32}\! \left(x \right)\\ \end{align*}\)