This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333650 #68 Aug 12 2022 19:23:06
%S A333650 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 A333650 160,64,1,54,228,401,497,510,368,128,1,88,467,944,1257,1412,1304,832,
%U A333650 256,1,143,949,2187,3172,3736,3879,3248,1856,512
%N A333650 Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.
%C A333650 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 A333650 Conjecture: For n > 1, T(n,2) = A000071(n+2).
%C A333650 A038622(n-1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.
%C A333650 Conjecture: T(n,n-1) = A339252(n-2). - _Peter Kagey_, Nov 21 2020
%C A333650 Conjecture: T(n,n-2) = A339254(n-3). - _Peter Luschny_, Nov 29 2020
%C A333650 Conjecture: T(n,n-3) = A339029(n-4). - _Peter Luschny_, Dec 01 2020
%C A333650 From _Peter Luschny_, Dec 01 2020: (Start)
%C A333650 The above conjectures can be summarized as follows:
%C A333650 T(2*n + k, n + k) = d_{n}(n + k - 1) for k >= 1 and 0 <= n <= 3, where
%C A333650 d_{0}(m) = 2^(m-1)*2;
%C A333650 d_{1}(m) = 2^(m-3)*(10 + 6*m);
%C A333650 d_{2}(m) = 2^(m-5)*(70 + 43*m + 9*m^2);
%C A333650 d_{3}(m) = 2^(m-7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)
%D A333650 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 25-27.
%H A333650 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 A333650 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 A333650 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 A333650 Peter Kagey, <a href="https://math.stackexchange.com/q/2949131/121988">Symmetric Brick Stacking</a>, Mathematics Stack Exchange, 2018.
%H A333650 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 A333650 Doron Zeilberger, <a href="/A333650/a333650.jpg">The 27 towers with 4 domino pieces</a>, illustration.
%F A333650 Row sums are given by A000244(n-1) = 3^(n-1).
%F A333650 T(n,1) = 1.
%F A333650 T(n,n) = 2^(n-1).
%e A333650 Table begins:
%e A333650 n\k| 1 2 3 4 5 6 7 8 9 10 11
%e A333650 ---+-----------------------------------------------------
%e A333650 1 | 1
%e A333650 2 | 1 2
%e A333650 3 | 1 4 4
%e A333650 4 | 1 7 11 8
%e A333650 5 | 1 12 24 28 16
%e A333650 6 | 1 20 52 70 68 32
%e A333650 7 | 1 33 110 168 193 160 64
%e A333650 8 | 1 54 228 401 497 510 368 128
%e A333650 9 | 1 88 467 944 1257 1412 1304 832 256
%e A333650 10 | 1 143 949 2187 3172 3736 3879 3248 1856 512
%e A333650 11 | 1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024
%e A333650 .
%e A333650 T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:
%e A333650 +-------+-------+ +-------+ +-------+
%e A333650 | | | | | | |
%e A333650 +---+---+---+---+, +---+---+---+---+, +-------+---+---+---+, and
%e A333650 | | | | | | | |
%e A333650 +-------+ +-------+-------+ +-------+-------+
%e A333650 +-------+
%e A333650 | |
%e A333650 +---+---+---+-------+.
%e A333650 | | |
%e A333650 +-------+-------+
%Y A333650 Cf. A000071 (col. 2), A339493 (col. 3), A000244, A038622, A168368, A264746, A320314, A339252, A339254, A339029, A339346, A339494.
%K A333650 nonn,tabl,hard
%O A333650 1,3
%A A333650 _Peter Kagey_, Mar 31 2020