Av(1243, 1342, 1432, 3142, 3241)
Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{4}-3 x^{3}+4 x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 168, 495, 1460, 4307, 12705, 37477, 110549, 326096, 961914, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(355 \left(\sqrt{331}\, \sqrt{3}-\frac{18867}{355}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+1696 \left(\sqrt{331}\, \sqrt{3}-\frac{993}{53}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3968 \sqrt{331}\, \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+436920 \left(\sqrt{3}-\frac{309 \sqrt{331}}{1655}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-2796288 \left(\sqrt{3}-\frac{21 \sqrt{331}}{331}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+709632 \sqrt{331}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}+15888 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)\right) \left(\frac{\sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(9 \sqrt{331}\, \sqrt{3}-249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1920}}{96}+\frac{\left(83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}-320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}-\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{6442647552}\\+\\\frac{\left(\left(\left(25728 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}} \sqrt{3}+16683 \,\mathrm{I} \left(\sqrt{3}-\frac{9 \sqrt{331}}{83}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+34944 \,\mathrm{I} \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+37224 \,\mathrm{I} \left(\sqrt{331}\, \sqrt{3}-\frac{83}{3}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-1588224 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-4494336 \,\mathrm{I}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}-15888 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)\right) \left(\frac{\mathrm{I} \sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}}{96}+\frac{\left(-83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}+320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}+\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{6442647552}\\+\\\frac{\left(\left(\left(\left(-355 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+18867 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+\left(-1696 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}+31776 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-3968 \sqrt{331}\, \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}-436920 \left(\sqrt{3}-\frac{309 \sqrt{331}}{1655}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+2796288 \left(\sqrt{3}-\frac{21 \sqrt{331}}{331}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-709632 \sqrt{331}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}+15888 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)\right) \left(-\frac{\sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(9 \sqrt{331}\, \sqrt{3}-249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1920}}{96}+\frac{\left(83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}-320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}-\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{6442647552}\\-\\\frac{47 \left(\left(\left(\frac{1072 \,\mathrm{I} \sqrt{3}\, 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}}{1551}+\frac{5561 \,\mathrm{I} \left(\sqrt{3}-\frac{9 \sqrt{331}}{83}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{12408}+\frac{1456 \,\mathrm{I} \sqrt{3}}{1551}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\mathrm{I} \,2^{\frac{1}{3}} \left(\sqrt{331}\, \sqrt{3}-\frac{83}{3}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-\frac{128 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}}{3}-\frac{17024 \,\mathrm{I}}{141}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}+\frac{662 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)}{1551}\right) \left(-\frac{\mathrm{I} \sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}}{96}+\frac{\left(-83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}+320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}+\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{8134656} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 83 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
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Copy 83 equations to clipboard:
\(\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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{18}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\
\end{align*}\)