Av(1243, 1324, 2314, 3142)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right)}{\left(x -1\right) \left(x^{3}-6 x^{2}+5 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 67, 222, 729, 2381, 7754, 25214, 81928, 266111, 864202, 2806273, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{3}-6 x^{2}+5 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}-3 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-6 a \! \left(n +1\right)+5 a \! \left(n +2\right)+1, \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \frac{\left(42 \,\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-5 \,\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}} \sqrt{3}+42 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+3 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-3\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{252}+\frac{\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+2\right)^{-n}}{3528}+\frac{\left(\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-84 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}-9 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(5 \,\mathrm{I} \sqrt{3}-3\right) \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{504}-\frac{\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+2\right)^{-n}}{3528}+1+\frac{\left(-42 \,\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}} \sqrt{3}+42 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+6 \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+2+\frac{\left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}-\frac{\mathrm{I} \sqrt{3}\, \left(756+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{504}\right)^{-n}}{3528}\)
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This specification was found using the strategy pack "Point Placements" and has 20 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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