Av(1243, 2134, 2413, 3142)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}{x^{7}-3 x^{6}-7 x^{5}+26 x^{4}-35 x^{3}+24 x^{2}-8 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 208, 664, 2122, 6786, 21701, 69381, 221769, 708745, 2264817, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-3 x^{6}-7 x^{5}+26 x^{4}-35 x^{3}+24 x^{2}-8 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 208\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+7 a \! \left(n +2\right)-26 a \! \left(n +3\right)+35 a \! \left(n +4\right)-24 a \! \left(n +5\right)+8 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 208\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+7 a \! \left(n +2\right)-26 a \! \left(n +3\right)+35 a \! \left(n +4\right)-24 a \! \left(n +5\right)+8 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +5}}{651142}+\frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +5}}{651142}+\frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +5}}{651142}+\frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +5}}{651142}+\frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +5}}{651142}+\frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +5}}{651142}+\frac{513043 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +5}}{651142}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{325571}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{325571}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{325571}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{325571}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{325571}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{325571}-\frac{603683 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +4}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{325571}-\frac{2187105 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +3}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{325571}+\frac{5252441 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +2}}{325571}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{651142}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{651142}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{651142}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{651142}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{651142}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{651142}-\frac{11148413 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +1}}{651142}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{325571}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{325571}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{325571}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{325571}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{325571}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{325571}-\frac{394111 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n -1}}{325571}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{651142}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{651142}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{651142}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{651142}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{651142}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{651142}+\frac{5250503 \mathit{RootOf} \left(Z^{7}-3 Z^{6}-7 Z^{5}+26 Z^{4}-35 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n}}{651142}\)
This specification was found using the strategy pack "Point Placements" and has 106 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
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\(\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_{27}\! \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_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \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_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{47}\! \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_{60}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{57}\! \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_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{70}\! \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_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{51}\! \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)\\
F_{67}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{86}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{86}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{98}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{105}\! \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_{64}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{94}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
\end{align*}\)