Av(1324, 1342, 2143, 2341, 2413, 3142)
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Generating Function
\(\displaystyle \frac{2 x^{6}-6 x^{5}+2 x^{4}-8 x^{3}+12 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 18, 50, 135, 359, 944, 2464, 6402, 16588, 42917, 110971, 286946, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)-2 x^{6}+6 x^{5}-2 x^{4}+8 x^{3}-12 x^{2}+6 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 135\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-9 a \! \left(n +2\right)+17 a \! \left(n +3\right)-17 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{8957 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{5}+9 Z^{4}-17 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{295}-\frac{54013 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{5}+9 Z^{4}-17 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{590}+\frac{27198 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{5}+9 Z^{4}-17 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{295}-\frac{12575 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{5}+9 Z^{4}-17 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{59}+\frac{45572 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{5}+9 Z^{4}-17 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{295}-\frac{18413 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{5}+9 Z^{4}-17 Z^{3}+17 Z^{2}-7 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{590}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 30 rules.

Found on July 23, 2021.

Finding the specification took 3 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_{16}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \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_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\ \end{align*}\)