Av(1243, 1432, 3412)
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Generating Function
\(\displaystyle -\frac{5 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 21, 72, 232, 707, 2066, 5858, 16257, 44428, 120076, 321919, 857942, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+5 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2 n +1, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{\left(-\sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(\sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+2 n -3 \,2^{n}+1\)
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This specification was found using the strategy pack "Point And Row Placements" and has 48 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_{19}\! \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_{19}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{28}\! \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_{11}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{10}\! \left(x \right) &= 0\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 2 F_{10}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22} \left(x \right)^{2} F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{19}\! \left(x \right) F_{30}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{19}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{19}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{19}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{19}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{19}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{19}\! \left(x \right) F_{43}\! \left(x \right)\\ \end{align*}\)