A333650 - OEIS

%I #68 Aug 12 2022 19:23:06

%S 1,1,2,1,4,4,1,7,11,8,1,12,24,28,16,1,20,52,70,68,32,1,33,110,168,193,

%T 160,64,1,54,228,401,497,510,368,128,1,88,467,944,1257,1412,1304,832,

%U 256,1,143,949,2187,3172,3736,3879,3248,1856,512

%N Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.

%C The towers must have a contiguous base of bricks, and each brick must be at least half supported below by another brick. The stacks do not need to be stable.

%C Conjecture: For n > 1, T(n,2) = A000071(n+2).

%C A038622(n-1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.

%C Conjecture: T(n,n-1) = A339252(n-2). - _Peter Kagey_, Nov 21 2020

%C Conjecture: T(n,n-2) = A339254(n-3). - _Peter Luschny_, Nov 29 2020

%C Conjecture: T(n,n-3) = A339029(n-4). - _Peter Luschny_, Dec 01 2020

%C From _Peter Luschny_, Dec 01 2020: (Start)

%C The above conjectures can be summarized as follows:

%C T(2*n + k, n + k) = d_{n}(n + k - 1) for k >= 1 and 0 <= n <= 3, where

%C d_{0}(m) = 2^(m-1)*2;

%C d_{1}(m) = 2^(m-3)*(10 + 6*m);

%C d_{2}(m) = 2^(m-5)*(70 + 43*m + 9*m^2);

%C d_{3}(m) = 2^(m-7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 25-27.

%H Peter Luschny, <a href="/A333650/b333650.txt">Table of n, a(n), for row(k) for k = 1..18</a> (the first 14 rows by Peter Kagey).

%H J. Bétréma and J.-G. Penaud, <a href="https://doi.org/10.1016/0304-3975(93)90304-C">Animaux et arbres guingois</a>, Theoretical Computer Science 117, 67-89, 1993.

%H D. Gouyou-Beauchamps and G. Viennot, <a href="https://doi.org/10.1016/0196-8858(88)90017-6">Equivalence of the two dimensional directed animals problem to a one-dimensional path problem</a>, Adv. in Appl. Math. 9(3), 334-357, 1988.

%H Peter Kagey, <a href="https://math.stackexchange.com/q/2949131/121988">Symmetric Brick Stacking</a>, Mathematics Stack Exchange, 2018.

%H Doron Zeilberger, <a href="https://arxiv.org/abs/1208.2258">The amazing 3^n theorem and its even more amazing proof</a>, arXiv:1208.2258 [math.CO], 2012.

%H Doron Zeilberger, <a href="/A333650/a333650.jpg">The 27 towers with 4 domino pieces</a>, illustration.

%F Row sums are given by A000244(n-1) = 3^(n-1).

%F T(n,1) = 1.

%F T(n,n) = 2^(n-1).

%e Table begins:

%e   n\k| 1   2    3    4    5    6     7     8    9   10   11

%e   ---+-----------------------------------------------------

%e    1 | 1

%e    2 | 1   2

%e    3 | 1   4    4

%e    4 | 1   7   11    8

%e    5 | 1  12   24   28   16

%e    6 | 1  20   52   70   68   32

%e    7 | 1  33  110  168  193  160    64

%e    8 | 1  54  228  401  497  510   368   128

%e    9 | 1  88  467  944 1257 1412  1304   832  256

%e   10 | 1 143  949 2187 3172 3736  3879  3248 1856  512

%e   11 | 1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024

%e .

%e T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:

%e +-------+-------+      +-------+                  +-------+

%e |       |       |      |       |                  |       |

%e +---+---+---+---+, +---+---+---+---+, +-------+---+---+---+, and

%e     |       |      |       |       |  |       |       |

%e     +-------+      +-------+-------+  +-------+-------+

%e +-------+

%e |       |

%e +---+---+---+-------+.

%e     |       |       |

%e     +-------+-------+

%Y Cf. A000071 (col. 2), A339493 (col. 3), A000244, A038622, A168368, A264746, A320314, A339252, A339254, A339029, A339346, A339494.

%K nonn,tabl,hard

%O 1,3

%A _Peter Kagey_, Mar 31 2020