I'm allowed to only use three blocks (these ones) and only use the corner pieces together, to build different shapes of height $3$ and variable width.
I'm tasked with finding a recursive pattern for the number of arrangements of width $1, 2, 3, 4. . .$.
I've drawn out the number of arrangements for the first 4 widths:(here) but I cannot seem to figure out the pattern.
I realize that:
- Starting with all the shapes of width $1$, you can combine shapes of width $3$ in $17$ different ways to build the shapes of width $4$,
- Starting with all the shapes of width $2$, you can make all the shapes of width $4$ (which is $21$)
- Starting with all the shapes of width $3$, it's the same as all the shapes as width $1$.
I do not see a specific (recursive) pattern here, can someone help me out? Thanks.
Using this information, you can define base cases $f(1) = 1, f(2) = 3, f(3) = 10$, and then the number of arrangements of width $n \ge 4$, $=f(n) = f(n-1) + 2f(n-2) + 5f(n-3)$.
