Av(1243, 1342, 2134, 2341, 4123)
Generating Function
\(\displaystyle \frac{-2 \left(x -\frac{1}{2}\right) \left(x -1\right)^{4} \sqrt{1-4 x}-4 x^{10}+10 x^{9}-4 x^{8}-6 x^{7}+6 x^{6}+4 x^{5}-11 x^{4}+16 x^{3}-14 x^{2}+6 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 159, 482, 1530, 5047, 17137, 59419, 209202, 745169, 2678823, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(2 x -1\right) \left(4 x^{10}-10 x^{9}+4 x^{8}+6 x^{7}-6 x^{6}-4 x^{5}+11 x^{4}-16 x^{3}+14 x^{2}-6 x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{19}-20 x^{18}+33 x^{17}-8 x^{16}-38 x^{15}+34 x^{14}+39 x^{13}-113 x^{12}+127 x^{11}-65 x^{10}-54 x^{9}+205 x^{8}-376 x^{7}+523 x^{6}-543 x^{5}+399 x^{4}-199 x^{3}+64 x^{2}-12 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 482\)
\(\displaystyle a \! \left(8\right) = 1530\)
\(\displaystyle a \! \left(9\right) = 5047\)
\(\displaystyle a \! \left(10\right) = 17137\)
\(\displaystyle a \! \left(11\right) = 59419\)
\(\displaystyle a \! \left(12\right) = 209202\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +4}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{n +4}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{n +4}-\frac{n^{4}-15 n^{3}+54 n^{2}-28 n -28}{2 \left(n +4\right)}, \quad n \geq 13\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 482\)
\(\displaystyle a \! \left(8\right) = 1530\)
\(\displaystyle a \! \left(9\right) = 5047\)
\(\displaystyle a \! \left(10\right) = 17137\)
\(\displaystyle a \! \left(11\right) = 59419\)
\(\displaystyle a \! \left(12\right) = 209202\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +4}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{n +4}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{n +4}-\frac{n^{4}-15 n^{3}+54 n^{2}-28 n -28}{2 \left(n +4\right)}, \quad n \geq 13\)
This specification was found using the strategy pack "Requirement Placements Tracked Fusion" and has 92 rules.
Found on July 23, 2021.Finding the specification took 11 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_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{10}\! \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_{6}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{30}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{43}\! \left(x \right) &= 0\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x , 1\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right) F_{81}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{62}\! \left(x \right)+F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{77}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{80}\! \left(x , y\right) F_{81}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= y x\\
F_{81}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{78}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= 2 F_{43}\! \left(x \right)+F_{83}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{84}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{43}\! \left(x \right)+F_{86}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{80}\! \left(x , y\right) F_{87}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{7}\! \left(x \right)+F_{85}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{89}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{74}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{91}\! \left(x , y\right) &= \frac{F_{70}\! \left(x , y\right) y -F_{70}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)