A123764 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2 which is flat, i.e., with all blocks in parallel position.

1, 2, 7, 24, 99, 416, 1854, 8407, 38970, 182742, 866442, 4140607, 19925401, 96430625, 469005432, 2290860538, 11232074043, 55255074216, 272634835875, 1348823736479, 6689314884962, 33247860759418, 165583649067958, 826170069700588, 4129098732200830

Offset

1,2

Comments

From Søren Eilers, Sep 12 2018: (Start)

The exponential growth is estimated to be 5.203 in Mølck Nilsson's MSc thesis. This puts an end to the speculation that it may be 5 at the end of the paper "Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length" by Durhuus and Eilers.

a(20)-a(25) follow from Rasmus Mølck Nilsson's extension of A319156 by transfer-matrix methods. (End)

Links

Table of n, a(n) for n=1..25.

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

B. Durhuus and S. Eilers, Combinatorial aspects of pyramids of one-dimensional pieces of fixed integer length. Drmota, Michael and Gittenberger, Bernhard. 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010, Vienna, Austria. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AM, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), pp.143-158, 2010, DMTCS Proceedings.

B. Durhuus and S. Eilers, On the entropy of LEGO, arXiv:math/0504039 [math.CO], 2005; Journal of Applied Mathematics & Computing 45 (2014) 433-448.

S. Eilers, A LEGO Counting problem, 2005.

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

R. Mølck Nilsson, On the number of flat LEGO structures [dead link]. MSc Thesis in mathematics, University of Copenhagen, 2016.

Index entry for sequences related to LEGO blocks

Formula

a(n) = (A319156(n)+A123765(n))/2. - Søren Eilers, Sep 12 2018

Crossrefs

Cf. A123765, A319156.

Sequence in context: A150436 A197553 A213950 * A150437 A150438 A326336

Adjacent sequences: A123761 A123762 A123763 * A123765 A123766 A123767

Keyword

nonn,hard,more

Author

Søren Eilers, Oct 29 2006

Extensions

a(20)-a(25) from Søren Eilers, Sep 12 2018

Status

approved