A123768 Number of ways to build a contiguous building with n LEGO blocks of size 1 X 2 on top of a fixed block of the same size so that the building is flat, i.e., with all blocks in parallel position.

3, 10, 39, 160, 693, 3071, 13953, 64335, 300749, 1420116, 6764691, 32453268, 156653477, 760139341, 3705381789, 18134664515, 89068494994, 438841011495

Offset

1,1

Comments

The base block is not counted among the n and must be the only block in the bottom layer of the building.

The minimum height of each building (counting the initial block) is 2 blocks for n=1 and n=2, and 3 blocks for n>2. The maximum height of each building is n+1 blocks for all n>=1. - Wesley Ivan Hurt, Nov 02 2014

Links

Table of n, a(n) for n=1..18.

M. Abrahamsen and S. Eilers, On the asymptotic enumeration of LEGO structures, Exper Math. 20 (2) (2011) 145-152.

B. Durhuus and S. Eilers, On the entropy of LEGO, arXiv:math/0504039 [math.CO], 2005.

S. Eilers, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.

Index entry for sequences related to LEGO blocks

Example

For n=1, there are 3 ways:
   ____     ____           ____
  |____|   |____|__     __|____|
  |____|,     |____|,  |____|
.
For n=2, there are 10 ways:
   ____       ____    ____      ____                ____
  |____|     |____|  |____|    |____|__          __|____|
  |____|   __|____|  |____|__     |____|__    __|____|
  |____|, |____|   ,    |____|,      |____|, |____|
   ____           ____       ____    ____
  |____|__     __|____|   __|____|  |____|__    ____ ____
   __|____|   |____|_    |____|        |____|  |____|____|
  |____|   ,    |____|,  |____|   ,    |____|,    |____|
.
From Wesley Ivan Hurt, Nov 02 2014

Crossrefs

Sequence in context: A219263 A050385 A296195 * A005750 A151068 A151069

Adjacent sequences: A123765 A123766 A123767 * A123769 A123770 A123771

Keyword

nonn,hard,more

Author

Søren Eilers, Oct 29 2006

Status

approved