Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 21345, 21354, 21435, 21453, 21534, 21543, 24135, 24153, 24513, 25134, 25143, 25413)
Generating Function
\(\displaystyle \frac{\left(2 x^{2}+2 x -1\right) \left(x^{5}+x^{4}-x^{3}+x^{2}+3 x -1\right)}{4 x^{7}+8 x^{6}-4 x^{5}-11 x^{4}+8 x^{3}+7 x^{2}-6 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 406, 1608, 6274, 24280, 93548, 359552, 1380034, 5292644, 20288734, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{7}+8 x^{6}-4 x^{5}-11 x^{4}+8 x^{3}+7 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x^{2}+2 x -1\right) \left(x^{5}+x^{4}-x^{3}+x^{2}+3 x -1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 406\)
\(\displaystyle a(7) = 1608\)
\(\displaystyle a{\left(n + 7 \right)} = - 4 a{\left(n \right)} - 8 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)} + 11 a{\left(n + 3 \right)} - 8 a{\left(n + 4 \right)} - 7 a{\left(n + 5 \right)} + 6 a{\left(n + 6 \right)}, \quad n \geq 8\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 406\)
\(\displaystyle a(7) = 1608\)
\(\displaystyle a{\left(n + 7 \right)} = - 4 a{\left(n \right)} - 8 a{\left(n + 1 \right)} + 4 a{\left(n + 2 \right)} + 11 a{\left(n + 3 \right)} - 8 a{\left(n + 4 \right)} - 7 a{\left(n + 5 \right)} + 6 a{\left(n + 6 \right)}, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle -\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +5}}{5390695}-\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +5}}{5390695}-\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +5}}{5390695}-\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +5}}{5390695}-\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +5}}{5390695}-\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +5}}{5390695}-\frac{3284852 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +5}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +4}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +4}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +4}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +4}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +4}}{5390695}-\frac{7833576 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +4}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +3}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +3}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +3}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +3}}{5390695}+\frac{374558 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +3}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +2}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +2}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +2}}{5390695}+\frac{9932447 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +2}}{5390695}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}}{1078139}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}}{1078139}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}}{1078139}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}}{1078139}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +1}}{1078139}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +1}}{1078139}-\frac{287183 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n -1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n -1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n -1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n -1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n -1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n -1}}{1078139}+\frac{347812 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n -1}}{1078139}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}}{5390695}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}}{5390695}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}}{5390695}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}}{5390695}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{5390695}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n}}{5390695}-\frac{6183888 \mathit{RootOf} \left(4 Z^{7}+8 Z^{6}-4 Z^{5}-11 Z^{4}+8 Z^{3}+7 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n}}{5390695}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 100 rules.
Finding the specification took 101 seconds.
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Copy 100 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{16}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{64}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{16}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{16}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{74}\! \left(x \right)+F_{81}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{16}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{74}\! \left(x \right)+F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{16}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{16}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{16}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{16}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{80}\! \left(x \right)\\
\end{align*}\)