Av(123, 132, 3214)
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Generating Function
\(\displaystyle -\frac{1}{x^{3}+x^{2}+x -1}\)
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Counting Sequence
1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right), \quad n \geq 3\)
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Explicit Closed Form
\(\displaystyle -\frac{\left(\left(\left(-\frac{27 i}{22}-\frac{9 \sqrt{3}}{22}\right) \sqrt{11}+\frac{5 i \sqrt{3}}{2}+\frac{5}{2}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-8+\left(\left(-\frac{9 i}{11}+\frac{3 \sqrt{3}}{11}\right) \sqrt{11}+i \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\left(\left(\left(-\frac{49 i}{33}+\frac{5 \sqrt{3}}{33}\right) \sqrt{11}+3 i \sqrt{3}-1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{29 i}{33}-\frac{7 \sqrt{3}}{33}\right) \sqrt{11}+i \sqrt{3}+1\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{2 i \sqrt{11}}{33}+6\right) \left(\frac{\left(\left(17 i+3 \sqrt{11}\right) \sqrt{3}-9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{i \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+4 \left(\frac{\left(\left(-17 i+3 \sqrt{11}\right) \sqrt{3}+9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{i \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+\left(\left(\left(-\frac{17 i}{33}-\frac{9 \sqrt{3}}{11}\right) \sqrt{11}+i \sqrt{3}+5\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-2+\left(-\frac{4 i}{33}+\frac{6 \sqrt{3}}{11}\right) \sqrt{11}\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{2 i \sqrt{11}}{33}+6\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}\right)}{96}\)
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This specification was found using the strategy pack "Point Placements" and has 15 rules.

Found on January 17, 2022.

Finding the specification took 1 seconds.

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