Av(1243, 1342, 3214)
Generating Function
\(\displaystyle \frac{\left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}{4 x^{6}-7 x^{5}+9 x^{4}-15 x^{3}+13 x^{2}-6 x +1}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 241, 768, 2415, 7587, 23905, 75507, 238759, 755088, 2387570, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{6}-7 x^{5}+9 x^{4}-15 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)-\left(x^{3}+2 x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 241\)
\(\displaystyle a \! \left(n +6\right) = -4 a \! \left(n \right)+7 a \! \left(n +1\right)-9 a \! \left(n +2\right)+15 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 241\)
\(\displaystyle a \! \left(n +6\right) = -4 a \! \left(n \right)+7 a \! \left(n +1\right)-9 a \! \left(n +2\right)+15 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{1367036 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +4}}{134023273}-\frac{250363 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +3}}{2851559}+\frac{4575648 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +2}}{134023273}-\frac{15453889 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +1}}{134023273}+\frac{2944759 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n -1}}{134023273}+\frac{27636497 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf} \left(4 Z^{6}-7 Z^{5}+9 Z^{4}-15 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n}}{134023273}+\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 88 rules.
Found on January 18, 2022.Finding the specification took 5 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_{20}\! \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_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \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_{60}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{61}\! \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_{64}\! \left(x \right) &= F_{48}\! \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_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{71}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{62}\! \left(x \right)\\
\end{align*}\)