Av(1342, 3142, 3241)
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Generating Function
\(\displaystyle \frac{-x \sqrt{1-4 x}-4 x^{2}+9 x -2}{2 x^{3}-8 x^{2}+10 x -2}\)
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Counting Sequence
1, 1, 2, 6, 21, 78, 298, 1157, 4539, 17936, 71251, 284188, 1137076, 4561093, 18333337, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-4 x^{2}+5 x -1\right) F \left(x \right)^{2}+\left(-1+4 x \right) \left(x -2\right) F \! \left(x \right)-1+4 x = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{3+n}+\frac{3 \left(7+3 n \right) a \! \left(3+n \right)}{3+n}+\frac{\left(27+17 n \right) a \! \left(n +1\right)}{3+n}-\frac{6 \left(7+4 n \right) a \! \left(n +2\right)}{3+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements" and has 17 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

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