###### 1324 Domino
Counting Sequence
1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, ...
Implicit Equation for the Generating Function
$$\displaystyle x^{4} F \left(x \right)^{3}+2 x^{2} \left(3 x +1\right) F \left(x \right)^{2}+\left(12 x^{2}-10 x +1\right) F \! \left(x \right)+8 x -1 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(n +1\right) = \frac{3 \left(3 n +5\right) \left(3 n +4\right) a \! \left(n \right)}{2 \left(n +3\right) \left(2 n +5\right)}, \quad n \geq 1$$
Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point $$(i, j)$$ represents how many permutations have value $$j$$ at index $$i$$ (darker = more).

### This specification was found using the strategy pack "Point Placements Tracked Fusion Expand Verified" and has 33 rules.

Found on February 02, 2022.

Finding the specification took 1872 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{1}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{2}\! \left(x \right) &= 1\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1} \left(x \right)^{2} F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= x\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{16}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right) F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{2}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{18}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right) F_{22}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{22}\! \left(x , y\right) &= -\frac{-y F_{23}\! \left(x , y\right)+F_{23}\! \left(x , 1\right)}{-1+y}\\ F_{23}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{1}\! \left(x \right) F_{23}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{23}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{1}\! \left(x \right) F_{32}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\ \end{align*}