A168368 -id:A168368 - cp's OEIS frontend

A333650 Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.

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, 160, 64, 1, 54, 228, 401, 497, 510, 368, 128, 1, 88, 467, 944, 1257, 1412, 1304, 832, 256, 1, 143, 949, 2187, 3172, 3736, 3879, 3248, 1856, 512

Offset: 1

Peter Kagey, Mar 31 2020

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.
Conjecture: For n > 1, T(n,2) = A000071(n+2).
A038622(n-1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.
Conjecture: T(n,n-1) = A339252(n-2). - Peter Kagey, Nov 21 2020
Conjecture: T(n,n-2) = A339254(n-3). - Peter Luschny, Nov 29 2020
Conjecture: T(n,n-3) = A339029(n-4). - Peter Luschny, Dec 01 2020
From Peter Luschny, Dec 01 2020: (Start)
The above conjectures can be summarized as follows:
T(2*n + k, n + k) = d_{n}(n + k - 1) for k >= 1 and 0 <= n <= 3, where
d_{0}(m) = 2^(m-1)*2;
d_{1}(m) = 2^(m-3)*(10 + 6*m);
d_{2}(m) = 2^(m-5)*(70 + 43*m + 9*m^2);
d_{3}(m) = 2^(m-7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)

			Table begins:
  n\k| 1   2    3    4    5    6     7     8    9   10   11
  ---+-----------------------------------------------------
   1 | 1
   2 | 1   2
   3 | 1   4    4
   4 | 1   7   11    8
   5 | 1  12   24   28   16
   6 | 1  20   52   70   68   32
   7 | 1  33  110  168  193  160    64
   8 | 1  54  228  401  497  510   368   128
   9 | 1  88  467  944 1257 1412  1304   832  256
  10 | 1 143  949 2187 3172 3736  3879  3248 1856  512
  11 | 1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024
.
T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:
+-------+-------+      +-------+                  +-------+
|       |       |      |       |                  |       |
+---+---+---+---+, +---+---+---+---+, +-------+---+---+---+, and
    |       |      |       |       |  |       |       |
    +-------+      +-------+-------+  +-------+-------+
+-------+
|       |
+---+---+---+-------+.
    |       |       |
    +-------+-------+

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

Peter Luschny, Table of n, a(n), for row(k) for k = 1..18 (the first 14 rows by Peter Kagey).
J. Bétréma and J.-G. Penaud, Animaux et arbres guingois, Theoretical Computer Science 117, 67-89, 1993.
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two dimensional directed animals problem to a one-dimensional path problem, Adv. in Appl. Math. 9(3), 334-357, 1988.
Peter Kagey, Symmetric Brick Stacking, Mathematics Stack Exchange, 2018.
Doron Zeilberger, The amazing 3^n theorem and its even more amazing proof, arXiv:1208.2258 [math.CO], 2012.
Doron Zeilberger, The 27 towers with 4 domino pieces, illustration.

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

Row sums are given by A000244(n-1) = 3^(n-1).
T(n,1) = 1.
T(n,n) = 2^(n-1).

A264746 a(n) is the number of domino towers with n bricks up to horizontal flipping.

Original entry on oeis.org

1, 2, 6, 15, 44, 126, 374, 1106, 3307, 9877, 29599, 88675, 265932, 797453, 2392089, 7175294, 21525097, 64572513, 193715253, 581137787, 1743406694, 5230197111, 15690571861, 47071649170, 141214890563, 423644479136, 1270933270658, 3812799252359, 11438397268254, 34315190174990

Offset: 1

Peter Kagey, Oct 10 2018

nonn

A domino tower is a stack of bricks, where (1) each row is offset from the preceding row by half of a brick, (2) the bottom row is contiguous, and (3) each brick is supported from below by at least half of a brick.

The number of domino towers with n bricks is given by 3^(n-1).

			For n=3, the a(3) = 6 domino towers are:
+-------+-------+-------+-------+
|       |       |       |       |
+-------+-------+-------+-------+
    +-------+
    |       |
+---+---+---+---+
|       |       |
+-------+-------+
+-------+-------+
|       |       |
+---+---+---+---+
    |       |
    +-------+
            +-------+
            |       |
+-------+---+---+---+
|       |       |
+-------+-------+
    +-------+
    |       |
+---+---+---+
|       |
+---+---+---+
    |       |
    +-------+
        +-------+
        |       |
    +---+---+---+
    |       |
+---+---+---+
|       |
+-------+

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000244, A168368, A320314.

a(n) = (3^(n-1) + A320314(n))/2

Terms a(20) and beyond from Andrew Howroyd, Mar 12 2021

A320314 a(n) is the number of symmetric domino towers with n bricks.

Original entry on oeis.org

1, 1, 3, 3, 7, 9, 19, 25, 53, 71, 149, 203, 423, 583, 1209, 1681, 3473, 4863, 10017, 14107, 28987, 41019, 84113, 119513, 244645, 348829, 712987, 1019731, 2081547, 2985097, 6086375, 8749185, 17820657, 25671983, 52241825, 75402907, 153316715, 221673707, 450393329, 652234089

Offset: 1

Peter Kagey, Oct 10 2018

nonn

A domino tower is a stack of bricks, where (1) each row is offset from the preceding row by half of a brick, (2) the bottom row is contiguous, and (3) each brick is supported from below by at least half of a brick.
The number of (not necessarily symmetric) domino towers with n blocks is given by 3^(n-1).
a(n) is odd for all n.
The not necessarily symmetric case is described in the Miklos Bona reference. Similar considerations lead to a decomposition of symmetric towers into half pyramids which are enumerated by the Motzkin numbers. - Andrew Howroyd, Mar 12 2021

			For n = 4, the a(4) = 3 symmetric stacks are
    +-------+
    |       |
+---+---+---+---+
|       |       |
+---+---+---+---+,
    |       |
    +-------+
+-------+       +-------+
|       |       |       |
+---+---+---+---+---+---+, and
    |       |       |
    +-------+-------+
+-------+-------+-------+-------+
|       |       |       |       |
+-------+-------+-------+-------+.

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Peter Kagey, Symmetric Brick Stacking, Mathematics Stack Exchange.

Cf. A000244, A001006, A168368, A264746.

seq(n)={my(h=(1 - x^2 - sqrt(1-2*x^2-3*x^4 + O(x^3*x^n)))/(2*x^2)); Vec((x + 2*x*h + h)/((1-x*h)*(1-h)))} \\ Andrew Howroyd, Mar 12 2021

G.f.: (x + 2*x^3*M(x^2) + x^2*M(x^2))/((1-x^3*M(x^2))*(1-x^2*M(x^2))) where M(x) is the g.f. of A001006. - Andrew Howroyd, Mar 12 2021

a(20)-a(40) from Andrew Howroyd, Oct 25 2018

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A333650 Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.

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A264746 a(n) is the number of domino towers with n bricks up to horizontal flipping.

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A320314 a(n) is the number of symmetric domino towers with n bricks.

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