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
Examples
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
| | | | | | | |
+-------+ +-------+-------+ +-------+-------+
+-------+
| |
+---+---+---+-------+.
| | |
+-------+-------+
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.
Comments