Av(12453, 12534, 12543, 13524, 21453, 21534, 21543)
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Counting Sequence
1, 1, 2, 6, 24, 113, 582, 3166, 17873, 103671, 613997, 3697198, 22565788, 139287889, 867973205, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}+x -3\right) F \left(x \right)^{3}-\left(x -1\right) \left(x -8\right) F \left(x \right)^{2}+\left(x -2\right) \left(x -3\right) F \! \left(x \right)+x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 582\)
\(\displaystyle a(7) = 3166\)
\(\displaystyle a(8) = 17873\)
\(\displaystyle a(9) = 103671\)
\(\displaystyle a(10) = 613997\)
\(\displaystyle a(11) = 3697198\)
\(\displaystyle a(12) = 22565788\)
\(\displaystyle a(13) = 139287889\)
\(\displaystyle a(14) = 867973205\)
\(\displaystyle a(15) = 5453031349\)
\(\displaystyle a(16) = 34501355345\)
\(\displaystyle a(17) = 219643102812\)
\(\displaystyle a{\left(n + 18 \right)} = \frac{8 \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} + \frac{\left(22039 n + 360426\right) a{\left(n + 17 \right)}}{1911 \left(n + 18\right)} - \frac{2 \left(34 n^{2} + 121 n + 97\right) a{\left(n + 1 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(151 n^{2} - 28675 n - 187270\right) a{\left(n + 6 \right)}}{637 \left(n + 17\right) \left(n + 18\right)} + \frac{2 \left(158 n^{2} + 1081 n + 1581\right) a{\left(n + 2 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{4 \left(217 n^{2} + 2244 n + 4889\right) a{\left(n + 3 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(895 n^{2} + 21937 n + 90380\right) a{\left(n + 5 \right)}}{637 \left(n + 17\right) \left(n + 18\right)} + \frac{\left(1921 n^{2} + 29291 n + 88728\right) a{\left(n + 4 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} + \frac{2 \left(2318 n^{2} + 2527 n - 108240\right) a{\left(n + 7 \right)}}{637 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(16857 n^{2} + 172017 n + 246182\right) a{\left(n + 8 \right)}}{637 \left(n + 17\right) \left(n + 18\right)} + \frac{4 \left(80231 n^{2} + 1630276 n + 8126661\right) a{\left(n + 11 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(82685 n^{2} + 2597813 n + 20398382\right) a{\left(n + 16 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} + \frac{2 \left(95714 n^{2} + 2795667 n + 20388088\right) a{\left(n + 15 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} + \frac{\left(119021 n^{2} + 1723905 n + 5616496\right) a{\left(n + 9 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(215269 n^{2} + 3797101 n + 16093396\right) a{\left(n + 10 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(312043 n^{2} + 8455049 n + 57095850\right) a{\left(n + 14 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} + \frac{\left(386061 n^{2} + 9633139 n + 59755052\right) a{\left(n + 13 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)} - \frac{\left(389081 n^{2} + 8828859 n + 49574608\right) a{\left(n + 12 \right)}}{1911 \left(n + 17\right) \left(n + 18\right)}, \quad n \geq 18\)
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This specification was found using the strategy pack "Point Placements Req Corrob" and has 32 rules.

Finding the specification took 994 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_{14}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{0}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 31 rules.

Finding the specification took 1965 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_{14}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{0}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ \end{align*}\)