Av(1234, 1432, 2143)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{7}+x^{6}-x^{5}+3 x^{3}+2 x^{2}+2 x -1}{x^{7}+x^{6}-x^{5}-x^{4}+2 x^{3}+x^{2}+3 x -1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 21, 72, 246, 845, 2901, 9955, 34165, 117254, 402409, 1381046, 4739681, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+x^{6}-x^{5}-x^{4}+2 x^{3}+x^{2}+3 x -1\right) F \! \left(x \right)-x^{7}-x^{6}+x^{5}-3 x^{3}-2 x^{2}-2 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 246\)
\(\displaystyle a \! \left(7\right) = 845\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)-a \! \left(n +3\right)+2 a \! \left(n +4\right)+a \! \left(n +5\right)+3 a \! \left(n +6\right), \quad n \geq 8\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +5}}{46213115}+\frac{922689 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n +5}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +4}}{46213115}+\frac{650133 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n +4}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +3}}{46213115}-\frac{155501 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n +3}}{46213115}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +2}}{9242623}+\frac{271720 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n +2}}{9242623}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n +1}}{46213115}+\frac{5145758 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n +1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n -1}}{46213115}+\frac{1758049 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n -1}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =6\right)^{-n}}{46213115}-\frac{685683 \mathit{RootOf} \left(Z^{7}+Z^{6}-Z^{5}-Z^{4}+2 Z^{3}+Z^{2}+3 Z -1, \mathit{index} =7\right)^{-n}}{46213115}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 103 rules.

Found on January 18, 2022.

Finding the specification took 11 seconds.

This tree is too big to show here. Click to view tree on new page.

Copy 103 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_{28}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= x^{2}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \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_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{63}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{75}\! \left(x \right)+F_{84}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{90}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{63}\! \left(x \right)\\ \end{align*}\)