Av(1243, 1342, 2143, 3214)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{2}}{x^{5}+6 x^{3}-8 x^{2}+5 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 202, 612, 1840, 5536, 16697, 50439, 152447, 460745, 1392319, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+6 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = -6 a \! \left(n +2\right)+8 a \! \left(n +3\right)-5 a \! \left(n +4\right)+a \! \left(n +5\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n \right) = -6 a \! \left(n +2\right)+8 a \! \left(n +3\right)-5 a \! \left(n +4\right)+a \! \left(n +5\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{4753216 \left(\left(\left(\left(\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{2561}-1\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}+\frac{3821}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{3065}{7683}+\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{2561}-1\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{11930 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}+\frac{4421}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{3521 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}-\frac{69739}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{40237}{15366}-\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}+\frac{3821}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(\frac{3521 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}-\frac{69739}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{10664 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}+\frac{14798}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{69833 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}-\frac{27782}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{3065}{7683}+\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{40237}{15366}-\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{69833 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}-\frac{27782}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{10597}{15366}-\frac{27782 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}+\left(\left(\left(-\frac{3221}{15366}-\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{2561}+\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{201}{2561}+\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{7683}-\frac{293 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{201 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}-\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{15366}-\frac{1609}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{201}{2561}+\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{7683}-\frac{293 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{79147 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{53426}{7683}-\frac{293 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{201 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{2561}+\frac{53426 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}-\frac{4477}{591}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{201 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}-\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{15366}-\frac{1609}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{201 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{2561}+\frac{53426 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}-\frac{4477}{591}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{1609 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{15366}-\frac{10597}{15366}-\frac{4477 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{591}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}+\left(\left(\left(1-\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{4}+\left(\left(\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{100 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}+\frac{1631}{197}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(6-\frac{12228 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+6 \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{600}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{46907}{15366}+\frac{39211 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{46907 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}+\frac{372262}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{50507 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}-\frac{9392}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{9392 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{230825}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}+\left(\left(\left(\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{2561}-1\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}+\frac{3821}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{3065}{7683}+\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}+\frac{3821}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{67277}{7683}+\frac{293 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{201 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{2561}-\frac{6438}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{3065}{7683}+\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{201 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{2561}-\frac{6438}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{1609 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}+\frac{30419}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}+\left(\left(\left(1-\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}+\left(\left(\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{100 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}+\frac{1631}{197}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{2462}{7683}-\frac{6749 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{2462 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{1609}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{2462 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{1609}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{1609 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}+\frac{30419}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}+\left(\left(\left(1-\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{3821}{15366}+\frac{4247 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}+\frac{66674}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}+\frac{3065}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{40237}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}+\left(\left(\left(\frac{173435}{15366}-\frac{55799 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{173435 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}-\frac{11792}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{173435 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}-\frac{11792}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{11792 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{1248569}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}+\left(\left(\left(-\frac{43636}{7683}+\frac{66619 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{43636 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{5209}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{43636 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{5209}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{5209 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}-\frac{351235}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}+\left(\left(\left(-\frac{3221}{15366}-\frac{3436 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}+\frac{3065}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{3221 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}+\frac{3065}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{40237}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}+\left(\left(\left(-1+\frac{2038 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{100 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}-\frac{1631}{197}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}+\left(\left(\left(\frac{3065}{7683}+\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}-\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{6438}{2561}-\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{15366}+\frac{67277 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{6438 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}+\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{7683}-\frac{30419}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{6438}{2561}-\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{15366}+\frac{67277 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{116177 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}-\frac{62890}{7683}+\frac{67277 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{6438 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{2561}-\frac{62890 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{311}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\frac{6438 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}+\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{7683}-\frac{30419}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{6438 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{2561}-\frac{62890 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}+\frac{311}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{30419 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{2}}{7683}-\frac{320303}{7683}+\frac{311 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}+\left(\left(\left(-\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}-\frac{66674}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{40237}{15366}-\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{3}-\left(\left(-\frac{100}{2561}+\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(-\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}+\frac{66674}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\frac{40237}{15366}+\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(\frac{3821 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}-\frac{66674}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{40237}{15366}-\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{2}+\left(\frac{38921 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{15366}-\frac{444026}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{32471 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}-\frac{266863}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{40237}{15366}-\frac{3065 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{3}+\left(-\frac{32471 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}-\frac{266863}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)-\frac{244835}{15366}+\frac{311 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}+\left(\left(\left(\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{100}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{100 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}+\frac{1631}{197}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{4}+\left(\left(6 \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)-\frac{600}{2561}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{600 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2561}+\frac{9786}{197}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{2}+\left(\left(-\frac{43982}{7683}+\frac{15995 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{43982 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}-\frac{1055551}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(-\frac{14081 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}-\frac{1957}{1182}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1957 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{1182}-\frac{297080}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}+\left(\left(\left(-\frac{46382}{7683}+\frac{138923 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{46382 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}-\frac{37807}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{46382 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{7683}-\frac{37807}{15366}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{37807 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{15366}-\frac{614825}{7683}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}-\frac{219669 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{5122}\right) \left(\left(\left(\left(\mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)+\frac{141}{464}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{141 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{464}-\frac{1321}{2784}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(\frac{141 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{464}-\frac{1321}{2784}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1321 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2784}+\frac{3}{116}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =1\right)+\left(\left(\frac{141 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{464}-\frac{1321}{2784}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)-\frac{1321 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2784}+\frac{3}{116}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =2\right)+\left(-\frac{1321 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{2784}+\frac{3}{116}\right) \mathit{RootOf} \left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =4\right)+\frac{3 \mathit{RootOf}\left(Z^{5}+6 Z^{3}-8 Z^{2}+5 Z -1, \mathit{index} =3\right)}{116}+\frac{13909}{2784}\right)}{5361607729}\)
This specification was found using the strategy pack "Point Placements" and has 67 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 67 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_{20}\! \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_{7}\! \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_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{33}\! \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_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{33}\! \left(x \right)+F_{48}\! \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_{57}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
\end{align*}\)