Av(1342, 1423, 3412)
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Generating Function
\(\displaystyle -\frac{4 x^{3}-7 x^{2}+5 x -1}{2 x^{4}-9 x^{3}+11 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 275, 991, 3563, 12800, 45976, 165141, 593184, 2130737, 7653715, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}-9 x^{3}+11 x^{2}-6 x +1\right) F \! \left(x \right)+4 x^{3}-7 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+9 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \frac{54 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-9 Z^{3}+11 Z^{2}-6 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{1423}+\frac{111 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-9 Z^{3}+11 Z^{2}-6 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{1423}+\frac{76 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-9 Z^{3}+11 Z^{2}-6 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{1423}+\frac{20 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-9 Z^{3}+11 Z^{2}-6 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{1423}\)
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This specification was found using the strategy pack "Point Placements" and has 27 rules.

Found on July 23, 2021.

Finding the specification took 5 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\ \end{align*}\)