Av(1234, 1324, 1342, 2143, 2314)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)}{x^{4}+2 x^{3}-4 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 166, 480, 1389, 4025, 11670, 33838, 98111, 284457, 824730, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{7 \left(\left(\left(\left(\mathrm{I} \sqrt{3}\, \sqrt{163}-\frac{49 \,\mathrm{I}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{200 \,\mathrm{I} \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{12700 \,\mathrm{I}}{21}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(-210 \,\mathrm{I} \sqrt{3}+\frac{270 \,\mathrm{I} \sqrt{163}}{7}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{6000 \,\mathrm{I} \sqrt{3}\, \left(\left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+44\right)}{7}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\frac{176040000}{7}+\left(\left(-\frac{6520 \sqrt{3}}{21}+\frac{440 \sqrt{163}}{7}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-\frac{32600 \sqrt{3}}{21}+\frac{200 \sqrt{163}}{7}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{326000 \sqrt{3}}{21}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}\right) \left(\left(\left(\left(\left(\left(-\frac{652}{3}-\frac{141 \,\mathrm{I} \sqrt{163}}{10}\right) \sqrt{3}+\frac{7009 \,\mathrm{I}}{30}+\frac{47 \sqrt{163}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{1793}{6}+\mathrm{I} \sqrt{163}\right) \sqrt{3}+\frac{2771 \,\mathrm{I}}{3}+\frac{713 \sqrt{163}}{6}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{55 \sqrt{163}}{3}+\frac{18745 \,\mathrm{I}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(\frac{17115 \,\mathrm{I}}{2}-\frac{1491 \sqrt{163}}{2}\right) \sqrt{3}-\frac{3015 \,\mathrm{I} \sqrt{163}}{2}-\frac{60147}{2}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(31785 \,\mathrm{I}+12675 \sqrt{163}\right) \sqrt{3}+315 \,\mathrm{I} \sqrt{163}-256725\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+48900 \sqrt{3}\, \left(\mathrm{I}+\frac{47 \sqrt{163}}{163}\right)\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(\left(\frac{331 \,\mathrm{I} \sqrt{163}}{3}+\frac{15974}{3}\right) \sqrt{3}-1141 \,\mathrm{I}-1228 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{595 \,\mathrm{I} \sqrt{163}}{3}+\frac{101875}{3}\right) \sqrt{3}+10595 \,\mathrm{I}-1225 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1400 \sqrt{3}\, \left(\mathrm{I} \sqrt{163}+\frac{7661}{7}\right)}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(102690 \,\mathrm{I}+20430 \sqrt{163}\right) \sqrt{3}-61290 \,\mathrm{I} \sqrt{163}-102690\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+363816000+\left(\left(-2493900 \,\mathrm{I}+69300 \sqrt{163}\right) \sqrt{3}+207900 \,\mathrm{I} \sqrt{163}-2493900\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(-\frac{\sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(18 \sqrt{163}\, \sqrt{3}-294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+26400}}{120}+\frac{\left(-49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}+\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\left(\left(\left(\left(\frac{652}{3}-\frac{141 \,\mathrm{I} \sqrt{163}}{10}\right) \sqrt{3}+\frac{7009 \,\mathrm{I}}{30}-\frac{47 \sqrt{163}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{1793}{6}+\mathrm{I} \sqrt{163}\right) \sqrt{3}+\frac{2771 \,\mathrm{I}}{3}-\frac{713 \sqrt{163}}{6}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{55 \sqrt{163}}{3}+\frac{18745 \,\mathrm{I}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(\frac{17115 \,\mathrm{I}}{2}+\frac{1491 \sqrt{163}}{2}\right) \sqrt{3}-\frac{3015 \,\mathrm{I} \sqrt{163}}{2}+\frac{60147}{2}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(31785 \,\mathrm{I}-12675 \sqrt{163}\right) \sqrt{3}+315 \,\mathrm{I} \sqrt{163}+256725\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+48900 \sqrt{3}\, \left(\mathrm{I}-\frac{47 \sqrt{163}}{163}\right)\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(\left(-\frac{331 \,\mathrm{I} \sqrt{163}}{3}+\frac{15974}{3}\right) \sqrt{3}+1141 \,\mathrm{I}-1228 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{595 \,\mathrm{I} \sqrt{163}}{3}+\frac{101875}{3}\right) \sqrt{3}-10595 \,\mathrm{I}-1225 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{1400 \sqrt{3}\, \left(\mathrm{I} \sqrt{163}-\frac{7661}{7}\right)}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(-102690 \,\mathrm{I}+20430 \sqrt{163}\right) \sqrt{3}+61290 \,\mathrm{I} \sqrt{163}-102690\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+363816000+\left(\left(2493900 \,\mathrm{I}+69300 \sqrt{163}\right) \sqrt{3}-207900 \,\mathrm{I} \sqrt{163}-2493900\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(18 \sqrt{163}\, \sqrt{3}-294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+26400}}{120}+\frac{\left(-49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}+\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\left(\left(\left(-\frac{577 \,\mathrm{I} \sqrt{163}\, \sqrt{3}}{20}+\frac{9291 \,\mathrm{I}}{20}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+14670 \,\mathrm{I}+\left(-2 \,\mathrm{I} \sqrt{163}\, \sqrt{3}+1956 \,\mathrm{I}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(6846 \,\mathrm{I} \sqrt{3}-1386 \,\mathrm{I} \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(34230 \,\mathrm{I} \sqrt{3}-630 \,\mathrm{I} \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+684600 \,\mathrm{I} \sqrt{3}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(-7172 \sqrt{3}+1152 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-21190 \sqrt{3}-810 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-130400 \sqrt{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(-40860 \sqrt{163}\, \sqrt{3}+205380\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-138600 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}+4987800 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+363816000\right) \left(-\frac{\mathrm{I} \sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(-18 \sqrt{163}\, \sqrt{3}+294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}-26400}}{120}+\frac{\left(49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}-\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+352080000 \left(\frac{\mathrm{I} \sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(-18 \sqrt{163}\, \sqrt{3}+294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}-26400}}{120}+\frac{\left(49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}-\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}\right)}{247920652800000000}\)
This specification was found using the strategy pack "Point Placements" and has 60 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 60 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_{14}\! \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_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \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_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right)\\
\end{align*}\)