Av(1234, 3421)
Generating Function
\(\displaystyle -\frac{3 x^{9}-10 x^{8}-4 x^{7}+45 x^{6}-64 x^{5}+82 x^{4}-60 x^{3}+29 x^{2}-8 x +1}{\left(x -1\right)^{9}}\)
Counting Sequence
1, 1, 2, 6, 22, 86, 321, 1085, 3266, 8797, 21478, 48206, 100728, 198046, 369617, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{9} F \! \left(x \right)+3 x^{9}-10 x^{8}-4 x^{7}+45 x^{6}-64 x^{5}+82 x^{4}-60 x^{3}+29 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 86\)
\(\displaystyle a \! \left(6\right) = 321\)
\(\displaystyle a \! \left(7\right) = 1085\)
\(\displaystyle a \! \left(8\right) = 3266\)
\(\displaystyle a \! \left(9\right) = 8797\)
\(\displaystyle a \! \left(n \right) = 4-\frac{969}{140} n +\frac{3709}{720} n^{2}-\frac{391}{2880} n^{4}-\frac{217}{180} n^{3}+\frac{23}{180} n^{5}+\frac{1}{2880} n^{8}-\frac{1}{1260} n^{7}-\frac{23}{1440} n^{6}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 86\)
\(\displaystyle a \! \left(6\right) = 321\)
\(\displaystyle a \! \left(7\right) = 1085\)
\(\displaystyle a \! \left(8\right) = 3266\)
\(\displaystyle a \! \left(9\right) = 8797\)
\(\displaystyle a \! \left(n \right) = 4-\frac{969}{140} n +\frac{3709}{720} n^{2}-\frac{391}{2880} n^{4}-\frac{217}{180} n^{3}+\frac{23}{180} n^{5}+\frac{1}{2880} n^{8}-\frac{1}{1260} n^{7}-\frac{23}{1440} n^{6}, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 4-\frac{969}{140} n +\frac{3709}{720} n^{2}-\frac{391}{2880} n^{4}-\frac{217}{180} n^{3}+\frac{23}{180} n^{5}+\frac{1}{2880} n^{8}-\frac{1}{1260} n^{7}-\\\frac{23}{1440} n^{6} & \text{otherwise} \end{array}\right.\)
Heatmap
To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 188 rules.
Found on April 28, 2021.Finding the specification took 6 seconds.
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Copy 188 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_{13}\! \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_{43}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \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_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= x\\
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_{28}\! \left(x \right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{33}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{13}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{13}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{16}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{13}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{44}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{13}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{23}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{13}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{13}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{33}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{13}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{13}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{13}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{16}\! \left(x \right)+F_{79}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{13}\! \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_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{13}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{84}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{13}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{13}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{91}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{13}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{13}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{96}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{91}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{13}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{101}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{102}\! \left(x \right)+F_{110}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{102}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{28}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{159}\! \left(x \right)+F_{16}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{124}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{121}\! \left(x \right)+F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{127}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{128}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{13}\! \left(x \right) F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{136}\! \left(x \right) &= 2 F_{16}\! \left(x \right)+F_{137}\! \left(x \right)+F_{145}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{13}\! \left(x \right) F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{13}\! \left(x \right) F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{142}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{137}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{13}\! \left(x \right) F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{13}\! \left(x \right) F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{148}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{102}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{13}\! \left(x \right) F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{151}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{152}\! \left(x \right)+F_{153}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{152}\! \left(x \right) &= 0\\
F_{153}\! \left(x \right) &= F_{13}\! \left(x \right) F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{13}\! \left(x \right) F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{13}\! \left(x \right) F_{158}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{13}\! \left(x \right) F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{163}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{145}\! \left(x \right)+F_{16}\! \left(x \right)+F_{164}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{13}\! \left(x \right) F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{164}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{13}\! \left(x \right) F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= 3 F_{16}\! \left(x \right)+F_{172}\! \left(x \right)+F_{173}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{172}\! \left(x \right) &= 0\\
F_{173}\! \left(x \right) &= F_{13}\! \left(x \right) F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{170}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{13}\! \left(x \right) F_{176}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{13}\! \left(x \right) F_{178}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{179}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{16}\! \left(x \right)+F_{181}\! \left(x \right)+F_{182}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{181}\! \left(x \right) &= 0\\
F_{182}\! \left(x \right) &= 0\\
F_{183}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{16}\! \left(x \right)+F_{173}\! \left(x \right)+F_{184}\! \left(x \right)+F_{185}\! \left(x \right)+F_{186}\! \left(x \right)\\
F_{184}\! \left(x \right) &= 0\\
F_{185}\! \left(x \right) &= 0\\
F_{186}\! \left(x \right) &= F_{13}\! \left(x \right) F_{187}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{179}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements" and has 204 rules.
Found on January 18, 2022.Finding the specification took 19 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 204 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_{45}\! \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_{37}\! \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_{23}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \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_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{24}\! \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_{34}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{198}\! \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_{49}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \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_{38}\! \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_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \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_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{71}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{177}\! \left(x \right)+F_{192}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{58}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \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_{82}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{15}\! \left(x \right)+F_{156}\! \left(x \right)+F_{171}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{64}\! \left(x \right)+F_{83}\! \left(x \right)+F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 0\\
F_{94}\! \left(x \right) &= 0\\
F_{95}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)+F_{121}\! \left(x \right)+F_{137}\! \left(x \right)+F_{15}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{101}\! \left(x \right) &= 0\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{106}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{113}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{108}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{102}\! \left(x \right)+F_{118}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{120}\! \left(x \right) &= 0\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{125}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{121}\! \left(x \right)+F_{134}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{136}\! \left(x \right) &= 0\\
F_{137}\! \left(x \right) &= 0\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{142}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{144}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{108}\! \left(x \right)+F_{138}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{160}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{127}\! \left(x \right)+F_{156}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{173}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{171}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{181}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{184}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{185}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{187}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{189}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{162}\! \left(x \right)+F_{177}\! \left(x \right)+F_{190}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{193}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{195}\! \left(x \right)\\
F_{195}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{192}\! \left(x \right)+F_{196}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{197}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{199}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{184}\! \left(x \right)+F_{200}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{201}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{198}\! \left(x \right)+F_{202}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{203}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{7}\! \left(x \right)\\
\end{align*}\)