Av(1243, 1324, 2314)
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Generating Function
\(\displaystyle \frac{-x -1+\sqrt{5 x^{2}-6 x +1}}{2 x \left(x -2\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 79, 311, 1265, 5275, 22431, 96900, 424068, 1876143, 8377299, 37704042, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x -2\right) F \left(x \right)^{2}+\left(x +1\right) F \! \left(x \right)-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = \frac{5 \left(1+n \right) a \! \left(n \right)}{2 \left(n +4\right)}-\frac{\left(25+16 n \right) a \! \left(1+n \right)}{2 \left(n +4\right)}+\frac{\left(34+13 n \right) a \! \left(n +2\right)}{2 n +8}, \quad n \geq 3\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 21 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= y x\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{12}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)