A383338 Square array read by antidiagonals, where the n-th row is the coordination sequence of a certain tiling with an n-dimensional analog of the X pentomino (or Greek cross), n >= 1.

1, 2, 1, 2, 4, 1, 2, 8, 8, 1, 2, 12, 26, 14, 1, 2, 16, 56, 76, 20, 1, 2, 20, 98, 244, 150, 28, 1, 2, 24, 152, 578, 632, 296, 38, 1, 2, 28, 218, 1138, 1882, 1680, 558, 48, 1, 2, 32, 296, 1984, 4492, 6424, 4336, 896, 60, 1, 2, 36, 386, 3176, 9230, 18908, 21782, 8688, 1422, 74, 1

Offset: 1

Author

Pontus von Brömssen, Apr 29 2025

Keywords

• nonn
• tabl

Comments

The tile consists of an n-dimensional central hypercube with one hypercube attached to each of its 2*n (n-1)-dimensional facets. n-dimensional space can be tiled with this tile by placing the centers of the tiles at integer points (x_1, ..., x_n) for which Sum_{j=1..n} j*x_j is divisible by 2*n+1. (See problem B6 of the 2019 Putnam competition). Two tiles are considered to be neighbors if they share an (n-1)-dimensional facet.

Examples

			Array begins:
  n\k| 0  1   2    3     4      5      6       7       8
  ---+--------------------------------------------------
  1  | 1  2   2    2     2      2      2       2       2
  2  | 1  4   8   12    16     20     24      28      32
  3  | 1  8  26   56    98    152    218     296     386
  4  | 1 14  76  244   578   1138   1984    3176    4774
  5  | 1 20 150  632  1882   4492   9230   17040   29042
  6  | 1 28 296 1680  6424  18908  46416   99904  194768
  7  | 1 38 558 4336 21782  80838 241730  616584 1393906
  8  | 1 48 896 8688 52896 232000 803232 2332896 5923776
		

Links

• Kiran S. Kedlaya, The 80th William Lowell Putnam Mathematical Competition, Dec 7 2019.
• Kiran S. Kedlaya, Solutions to the 80th William Lowell Putnam Mathematical Competition, Dec 7 2019.
• Yusuke Nakamura, Ryotaro Sakamoto, Takafumi Mase, and Junichi Nakagawa, Coordination sequences of crystals are of quasi-polynomial type, Acta Crystallographica A 77 (2021), 138-148.
• Eric Weisstein's World of Mathematics, Greek Cross.
• Index entries for sequences related to coordination sequences.

Crossrefs

Rows: A040000 (n=1), A008574 (n=2), A005897 (n=3; empirically).

Cf. A007980.

Formula

T(n,0) = 1.

T(n,1) = 2*A007980(n-1).

T(1,k) = A040000(k).

T(2,k) = A008574(k).

Empirically (do these formulas follow from the results of Nakamura et al.?):

T(3,k) = A005897(k).

T(4,k) = 10*k^3 - 7*k^2 + 13*k - 2 for k >= 1.

T(5,k) = (22/3)*k^4 - 4*k^3 + (50/3)*k^2 - 2*k + 2 for k >= 1.

T(6,k) = (32/5)*k^5 - 7*k^4 + 28*k^3 - 11*k^2 + (58/5)*k for k >= 1.

T(7,k) = (304/45)*k^6 - (284/15)*k^5 + (1237/18)*k^4 - 86*k^3 + (8777/90)*k^2 - (601/15)*k + 10 for k >= 1.