Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 24153, 25134, 25143, 31245, 31254, 32145, 32154)
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Generating Function
\(\displaystyle \frac{4 x^{4}+8 x^{3}+2 x^{2}+x -1}{4 x^{4}+8 x^{3}+2 x^{2}+2 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 80, 264, 904, 3072, 10384, 35200, 119360, 404480, 1370816, 4646272, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{4}+8 x^{3}+2 x^{2}+2 x -1\right) F \! \left(x \right)-4 x^{4}-8 x^{3}-2 x^{2}-x +1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = 4 a{\left(n \right)} + 8 a{\left(n + 1 \right)} + 2 a{\left(n + 2 \right)} + 2 a{\left(n + 3 \right)}, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-7116 \sqrt{2}\, \left(\sqrt{3}-\frac{259 \sqrt{593}}{2372}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(2116 i \sqrt{3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4232}-40917 \sqrt{2}\, \sqrt{3}-690 \sqrt{2}\, \sqrt{593}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4 i \sqrt{3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4232}\, \left(1426+\left(3 \sqrt{593}\, \sqrt{3}-62\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}\right)\right) \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+4 i \sqrt{3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4232}\, \sqrt{2}\, \left(\left(620 \sqrt{3}-90 \sqrt{593}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+5290 \sqrt{3}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-137011 \sqrt{3}\right)\right) \left(\frac{\left(-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+2116\right) \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}}{876024}-\frac{i \sqrt{18 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+276 \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}-3174 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-25392}}{276}+\frac{\sqrt{\left(136390+13639 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-2372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+153587 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+4078061}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}}{97336}-\frac{1}{2}\right)^{-n}}{1038964464}\\+\\\frac{\left(\left(-7116 \sqrt{2}\, \left(\sqrt{3}-\frac{259 \sqrt{593}}{2372}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-2116 i \sqrt{3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4232}-40917 \sqrt{2}\, \sqrt{3}-690 \sqrt{2}\, \sqrt{593}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4 i \sqrt{3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4232}\, \left(1426+\left(3 \sqrt{593}\, \sqrt{3}-62\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}\right)\right) \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-4 i \sqrt{3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-4232}\, \sqrt{2}\, \left(\left(620 \sqrt{3}-90 \sqrt{593}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+5290 \sqrt{3}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-137011 \sqrt{3}\right)\right) \left(\frac{\left(-529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+2116\right) \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}}{876024}+\frac{i \sqrt{18 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}-372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+276 \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}-3174 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-25392}}{276}+\frac{\sqrt{\left(136390+13639 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-2372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+153587 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+4078061}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}}{97336}-\frac{1}{2}\right)^{-n}}{1038964464}\\+\\\frac{\left(\left(7116 \sqrt{2}\, \left(\sqrt{3}-\frac{259 \sqrt{593}}{2372}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(40917 \sqrt{2}\, \sqrt{3}+690 \sqrt{2}\, \sqrt{593}-2116 \sqrt{-3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4232}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \sqrt{-3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4232}\, \left(1426+\left(3 \sqrt{593}\, \sqrt{3}-62\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}\right)\right) \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+4 \sqrt{-3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4232}\, \sqrt{2}\, \left(\left(620 \sqrt{3}-90 \sqrt{593}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+5290 \sqrt{3}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-137011 \sqrt{3}\right)\right) \left(\frac{\left(529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}-2116\right) \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}}{876024}-\frac{\sqrt{\left(136390+13639 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-2372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+153587 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+4078061}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}}{97336}-\frac{\sqrt{-18 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+276 \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}+372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+3174 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+25392}}{276}-\frac{1}{2}\right)^{-n}}{1038964464}\\-\\\frac{5 \left(\frac{\left(529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}-2116\right) \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}}{876024}-\frac{\sqrt{\left(136390+13639 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-2372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+153587 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+4078061}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}}{97336}+\frac{\sqrt{-18 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+276 \sqrt{\left(690+69 \sqrt{593}\, \sqrt{3}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-12 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+777 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+20631}+372 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+3174 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+25392}}{276}-\frac{1}{2}\right)^{-n} \left(\left(-\frac{1779 \sqrt{2}\, \left(\sqrt{3}-\frac{259 \sqrt{593}}{2372}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}}{5290}+\left(-\frac{1779 \sqrt{2}\, \sqrt{3}}{920}-\frac{3 \sqrt{2}\, \sqrt{593}}{92}-\frac{\sqrt{-3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4232}}{10}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{\sqrt{-3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4232}\, \left(1426+\left(3 \sqrt{593}\, \sqrt{3}-62\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}\right)}{5290}\right) \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+\frac{\sqrt{-3 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+23 \sqrt{2}\, \sqrt{3}\, \sqrt{\left(46 \sqrt{593}\, \sqrt{3}+460\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{593}\, \sqrt{3}+518 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+13754}+62 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+529 \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}+4232}\, \sqrt{2}\, \left(\left(620 \sqrt{3}-90 \sqrt{593}\right) \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{2}{3}}+5290 \sqrt{3}\, \left(62+3 \sqrt{593}\, \sqrt{3}\right)^{\frac{1}{3}}-137011 \sqrt{3}\right)}{5290}\right)}{245502} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 91 rules.

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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_{23}\! \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_{16}\! \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_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= x^{2}\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{54}\! \left(x \right) &= 0\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 0\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 0\\ F_{66}\! \left(x \right) &= 0\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{70}\! \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_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{59}\! \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_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{78}\! \left(x \right)\\ \end{align*}\)