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

1,3

Comments

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)

References

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

Links

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.

Formula

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

T(n,1) = 1.

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

Example

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
    |       |      |       |       |  |       |       |
    +-------+      +-------+-------+  +-------+-------+
+-------+
|       |
+---+---+---+-------+.
    |       |       |
    +-------+-------+

Crossrefs

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

Sequence in context: A008572 A209150 A209145 * A214984 A118976 A210235

Adjacent sequences: A333647 A333648 A333649 * A333651 A333652 A333653

Keyword

nonn,tabl,hard

Author

Peter Kagey, Mar 31 2020

Status

approved