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Suppose you stack $n$ LEGO bricks ($2 \times 1$) in a plane, where

  • The base is contiguous
  • Each level is offset from the level below it by one stud.
  • Bricks are only stacked on top of other bricks, not below.

It turns out that there are exactly $3^{n-1}$ such stacks. (See here beginning on page 25.)


Question

How many such stacks are left-right symmetric? By my brute force program:

n  | # symmetric stacks
---+-------------------
1  | 1
2  | 1
3  | 3
4  | 3
5  | 7
6  | 9
7  | 19
8  | 25
9  | 53
10 | 71
11 | 149
12 | 203
13 | 423
14 | 583
15 | 1209

And by a parity argument, there are an odd number of such stacks for each value of $n$.


Examples

For example, the following three stacks of four bricks are legal: enter image description here enter image description here enter image description here


Non-Examples

The following three stacks are not legal because they violate the three conditions above: in the first, the base is not contiguous; in the second, the levels are not offset; and in the third, the second brick in the second row doesn't have any bricks below it. enter image description here enter image description here enter image description here

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  • $\begingroup$ This is extremely similar to a Project Euler Problem $\endgroup$ Oct 9, 2018 at 21:12
  • $\begingroup$ My other question is loosely related. $\endgroup$ Oct 9, 2018 at 21:13
  • $\begingroup$ @RushabhMehta, can you find which problem? Perhaps Problem 215? $\endgroup$ Oct 9, 2018 at 21:13
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    $\begingroup$ Oops, this is not nearly as similar as I remember it to be. My bad. Here is the problem. $\endgroup$ Oct 9, 2018 at 21:16
  • $\begingroup$ I've submitted this to the OEIS as A320314. Right now it's a draft, but I expect it will be added as a sequence within a few days. $\endgroup$ Oct 10, 2018 at 20:23

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