Av(1342, 1423, 1432, 2314, 2413)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{2}+2 x -1\right)}{x^{4}-3 x^{2}+4 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 59, 181, 553, 1688, 5152, 15725, 47997, 146501, 447165, 1364882, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-3 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = 3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle -\frac{29 \left(\left(\left(\left(\left(\left(\frac{13 \sqrt{3}}{1769}-\frac{387 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{151}{1769}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{653 \sqrt{3}}{5307}+\frac{55 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{2945}{1769}-\frac{3 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{498}{1769}-\frac{30 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(\frac{104 \sqrt{3}}{1769}+\frac{2568 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{3336}{1769}-\frac{204 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{1288 \sqrt{3}}{1769}-\frac{1776 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{17336}{1769}+\frac{124 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{5820}{1769}+\frac{1224 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\left(\frac{528 \,\mathrm{I}}{1769}+\frac{156 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{41 \,\mathrm{I} \sqrt{3}}{29}-\frac{27}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{250 \,\mathrm{I}}{1769}-\frac{306 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{17 \,\mathrm{I} \sqrt{3}}{29}+\frac{69}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{156 \,\mathrm{I} \sqrt{61}}{1769}+\frac{198}{29}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(-\frac{4140 \,\mathrm{I}}{1769}-\frac{1380 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{324 \,\mathrm{I} \sqrt{3}}{29}+\frac{324}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1584}{29}+\left(\left(\frac{2268 \,\mathrm{I}}{1769}-\frac{756 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{156 \,\mathrm{I} \sqrt{3}}{29}+\frac{156}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}-\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}-\frac{\sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}\right)^{-n}+\left(\left(\left(\left(\left(-\frac{13 \sqrt{3}}{1769}-\frac{387 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{151}{1769}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{653 \sqrt{3}}{5307}+\frac{55 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{2945}{1769}-\frac{3 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{498}{1769}-\frac{30 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(-\frac{104 \sqrt{3}}{1769}+\frac{2568 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{3336}{1769}-\frac{204 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{1288 \sqrt{3}}{1769}-\frac{1776 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{17336}{1769}+\frac{124 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{5820}{1769}+\frac{1224 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\left(-\frac{528 \,\mathrm{I}}{1769}+\frac{156 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{41 \,\mathrm{I} \sqrt{3}}{29}-\frac{27}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{250 \,\mathrm{I}}{1769}-\frac{306 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{17 \,\mathrm{I} \sqrt{3}}{29}+\frac{69}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{156 \,\mathrm{I} \sqrt{61}}{1769}+\frac{198}{29}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(\frac{4140 \,\mathrm{I}}{1769}-\frac{1380 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{324 \,\mathrm{I} \sqrt{3}}{29}+\frac{324}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1584}{29}+\left(\left(-\frac{2268 \,\mathrm{I}}{1769}-\frac{756 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{156 \,\mathrm{I} \sqrt{3}}{29}+\frac{156}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}-\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}+\frac{\sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}\right)^{-n}+\left(\left(\left(\left(-\frac{386 \,\mathrm{I} \sqrt{61}}{1769}+\frac{74 \,\mathrm{I} \sqrt{3}}{87}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\frac{1394 \,\mathrm{I} \sqrt{61}}{5307}+\frac{106 \,\mathrm{I} \sqrt{3}}{87}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{148 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\frac{3864 \,\mathrm{I} \sqrt{61}}{1769}-\frac{264 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\frac{2568 \,\mathrm{I} \sqrt{61}}{1769}-\frac{200 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{504 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(-\frac{312 \sqrt{61}\, \sqrt{3}}{1769}+\frac{90}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{108}{29}-\frac{852 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}}{1769}+\frac{186 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}}{29}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-\frac{648}{29}+\frac{2760 \sqrt{61}\, \sqrt{3}}{1769}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1584}{29}-\frac{312 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}}{29}+\frac{1512 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}}{1769}\right) \left(\frac{\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}-\frac{\mathrm{I} \sqrt{3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-36+27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}-\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}+\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}\right)^{-n}-\frac{432 \left(\frac{\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\mathrm{I} \sqrt{3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-36+27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}-\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}+\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}\right)^{-n}}{29}\right) \left(\left(\left(\left(-\frac{18 \,\mathrm{I} \sqrt{61}}{61}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(3 \,\mathrm{I} \sqrt{3}-\frac{40 \,\mathrm{I} \sqrt{61}}{61}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{6 \,\mathrm{I} \sqrt{61}}{61}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-24 \,\mathrm{I} \sqrt{3}+\frac{342 \,\mathrm{I} \sqrt{61}}{61}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-12 \,\mathrm{I} \sqrt{3}+\frac{150 \,\mathrm{I} \sqrt{61}}{61}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{18 \,\mathrm{I} \sqrt{61}}{61}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+324+\left(\left(\frac{39 \sqrt{61}\, \sqrt{3}}{61}-9\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{15 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}}{61}-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+18\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}\right)}{559872}\)
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This specification was found using the strategy pack "Point Placements" and has 31 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ \end{align*}\)