Av(1243, 1342, 2143, 3412)
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Generating Function
\(\displaystyle \frac{x^{6}-4 x^{5}+10 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{\left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 196, 583, 1702, 4906, 14010, 39716, 111912, 313741, 875690, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{6}+4 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 196\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-n, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-5 n +26\right) \sqrt{5}+15 n -50\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(5 n -26\right) \sqrt{5}+15 n -50\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-n +2 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Row And Col Placements" and has 30 rules.

Found on January 17, 2022.

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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ 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_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{0}\! \left(x \right) F_{13}\! \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_{3}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{17}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{24}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ \end{align*}\)