Av(1342, 1432, 2413, 3241, 3412)
Generating Function
\(\displaystyle \frac{x^{6}-3 x^{5}+3 x^{4}-2 x^{3}+5 x^{2}-4 x +1}{\left(x^{4}-x^{3}+x^{2}-3 x +1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 154, 425, 1163, 3170, 8626, 23455, 63756, 173280, 470925, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-x^{3}+x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{6}+3 x^{5}-3 x^{4}+2 x^{3}-5 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 154\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right)+n +2, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 154\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+3 a \! \left(n +3\right)+n +2, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-10976 \,\mathrm{I} \sqrt{3}\, 2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-59339 \,\mathrm{I} \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{9 \sqrt{1099}}{173}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-79072 \,\mathrm{I} \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}-79884 \,\mathrm{I} \left(\sqrt{1099}\, \sqrt{3}-\frac{173}{3}\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+852096 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+8050560 \,\mathrm{I}\right) \sqrt{24 \sqrt{3}\, \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(-9 \sqrt{1099}\, \sqrt{3}+519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+480}-425392128+\left(1327592 \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{8985 \sqrt{1099}}{165949}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-914368 \,2^{\frac{2}{3}} \left(\sqrt{3}-\frac{951 \sqrt{1099}}{14287}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+2813440 \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\right) \left(-\frac{\mathrm{I} \sqrt{24 \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(-9 \sqrt{1099}\, \sqrt{3}+519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+480}}{48}+\frac{\left(-173 \,2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-32 \left(692+12 \sqrt{1099}\, \sqrt{3}\right)^{\frac{1}{3}}+80\right) \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}}{48384}+\frac{\sqrt{-17584 \,2^{\frac{2}{3}} \left(\sqrt{1099}\, \sqrt{3}-61\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-228592+\left(-5495 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+363769 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}{5376}+\frac{1}{4}\right)^{-n}}{850784256}\\+\\\frac{\left(\left(\left(10976 \,\mathrm{I} \sqrt{3}\, 2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+59339 \,\mathrm{I} \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{9 \sqrt{1099}}{173}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+79072 \,\mathrm{I} \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+79884 \,\mathrm{I} \left(\sqrt{1099}\, \sqrt{3}-\frac{173}{3}\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-852096 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-8050560 \,\mathrm{I}\right) \sqrt{24 \sqrt{3}\, \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(-9 \sqrt{1099}\, \sqrt{3}+519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+480}-425392128+\left(1327592 \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{8985 \sqrt{1099}}{165949}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-914368 \,2^{\frac{2}{3}} \left(\sqrt{3}-\frac{951 \sqrt{1099}}{14287}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+2813440 \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\right) \left(\frac{\mathrm{I} \sqrt{24 \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(-9 \sqrt{1099}\, \sqrt{3}+519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+480}}{48}+\frac{\left(-173 \,2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-32 \left(692+12 \sqrt{1099}\, \sqrt{3}\right)^{\frac{1}{3}}+80\right) \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}}{48384}+\frac{\sqrt{-17584 \,2^{\frac{2}{3}} \left(\sqrt{1099}\, \sqrt{3}-61\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-228592+\left(-5495 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+363769 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}{5376}+\frac{1}{4}\right)^{-n}}{850784256}\\+\\\frac{\left(\left(\left(\left(-833 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+49455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+\left(532 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}-39564 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+1568 \sqrt{1099}\, \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}-1200108 \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{803 \sqrt{1099}}{14287}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-1846320 \left(\sqrt{3}-\frac{323 \sqrt{1099}}{5495}\right) 2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-99456 \sqrt{1099}\right) \sqrt{24 \sqrt{3}\, \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(-9 \sqrt{1099}\, \sqrt{3}+519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+480}-425392128+\left(-1327592 \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{8985 \sqrt{1099}}{165949}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+914368 \,2^{\frac{2}{3}} \left(\sqrt{3}-\frac{951 \sqrt{1099}}{14287}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-2813440 \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\right) \left(-\frac{\sqrt{24 \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(9 \sqrt{1099}\, \sqrt{3}-519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-480}}{48}+\frac{\left(173 \,2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+32 \left(692+12 \sqrt{1099}\, \sqrt{3}\right)^{\frac{1}{3}}-80\right) \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}}{48384}-\frac{\sqrt{-17584 \,2^{\frac{2}{3}} \left(\sqrt{1099}\, \sqrt{3}-61\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-228592+\left(-5495 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+363769 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}{5376}+\frac{1}{4}\right)^{-n}}{850784256}\\+\\\frac{\left(\left(\left(\left(833 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}-49455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+\left(-532 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+39564 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-1568 \sqrt{1099}\, \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+1200108 \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{803 \sqrt{1099}}{14287}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+1846320 \left(\sqrt{3}-\frac{323 \sqrt{1099}}{5495}\right) 2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+99456 \sqrt{1099}\right) \sqrt{24 \sqrt{3}\, \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(-9 \sqrt{1099}\, \sqrt{3}+519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}+480}-425392128+\left(-1327592 \,2^{\frac{1}{3}} \left(\sqrt{3}-\frac{8985 \sqrt{1099}}{165949}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+914368 \,2^{\frac{2}{3}} \left(\sqrt{3}-\frac{951 \sqrt{1099}}{14287}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-2813440 \sqrt{3}\right) \sqrt{\left(-16 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+976 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-208+\left(-5 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+331 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\right) \left(\frac{\sqrt{24 \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}+\left(9 \sqrt{1099}\, \sqrt{3}-519\right) 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}-96 \,2^{\frac{2}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-480}}{48}+\frac{\left(173 \,2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}+32 \left(692+12 \sqrt{1099}\, \sqrt{3}\right)^{\frac{1}{3}}-80\right) \sqrt{\left(-48 \sqrt{1099}\, 2^{\frac{2}{3}} \sqrt{3}+2928 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-624+\left(-15 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+993 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}}{48384}-\frac{\sqrt{-17584 \,2^{\frac{2}{3}} \left(\sqrt{1099}\, \sqrt{3}-61\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{1}{3}}-228592+\left(-5495 \sqrt{1099}\, \sqrt{3}\, 2^{\frac{1}{3}}+363769 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{1099}\, \sqrt{3}+173\right)^{\frac{2}{3}}}{5376}+\frac{1}{4}\right)^{-n}}{850784256}\\-n +2 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 84 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
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Copy 84 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \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_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \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_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{10}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{10}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{10}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{10}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{10}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{10}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{71}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{10}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{10}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{10}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{10}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{80}\! \left(x \right)\\
\end{align*}\)