Av(1243, 1342, 2143, 2413, 4123)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{4}-3 x^{3}+4 x^{2}-4 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 168, 495, 1460, 4307, 12705, 37477, 110549, 326096, 961914, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 5\)
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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 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-\frac{993}{53}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3968 \sqrt{331}\, \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \,2^{\frac{2}{3}} \left(\sqrt{3}-\frac{21 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+709632 \sqrt{331}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}} \sqrt{3}\, \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+18867 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+\left(-1696 \sqrt{331}\, \sqrt{3}\, 2^{\frac{2}{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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \,2^{\frac{2}{3}} \left(\sqrt{3}-\frac{21 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-709632 \sqrt{331}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\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}}-\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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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}\, \sqrt{3}\, 2^{\frac{2}{3}}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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 \sqrt{331}\, \sqrt{3}\, 2^{\frac{1}{3}}+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.\)
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This specification was found using the strategy pack "Point Placements" and has 114 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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Copy 114 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{36}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ 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_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \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_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 0\\ F_{46}\! \left(x \right) &= F_{4}\! \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_{4}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \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_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{60}\! \left(x \right)+F_{89}\! \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_{69}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{60}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{74}\! \left(x \right)+F_{85}\! \left(x \right)+F_{87}\! \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_{79}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{28}\! \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_{66}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 2 F_{45}\! \left(x \right)+F_{74}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{92}\! \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_{97}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{109}\! \left(x \right)\\ \end{align*}\)