A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2*k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n*(n + 1)/2, read by rows.

1, 1, 3, 1, 9, 24, 18, 1, 18, 126, 434, 762, 630, 187, 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135, 1, 45, 915, 11127, 90270, 515970, 2139120, 6523428, 14683401, 24256853, 28975770, 24383838, 13860321, 4966929, 989970, 81462, 1, 63

Offset

1,3

Links

R. J. Mathar, Table of n, a(n) for n = 1..575

J. A. De Loera, J. Rambau, F. Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511.

R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.

Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004.

Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.

Wikipedia, Lozenge.

Formula

T(n,2) = 3*(n-1)*(n-2)*(3*n^2+3*n-4)/8 . - R. J. Mathar, May 24 2016

T(n,3) = (n-2)*(9*n^5-9*n^4-81*n^3+81*n^2+160*n-192)/16. - Greg Dresden, Jul 03 2019

Conjecture: T(n,4) = 3*(n-2)*(n-3)*(9*n^6+9*n^5-135*n^4-81*n^3+670*n^2+104*n-1216)/128. - Greg Dresden, Jul 03 2019

Conjecture: T(n,5) = 3*(n-3)*(n+3)* (27*n^8 -135*n^7 -387*n^6 +2835*n^5 -168*n^4 -18732*n^3 +19568*n^2 +36992*n -56320)/1280. - R. J. Mathar, Jul 07 2019

From Petros Hadjicostas, Sep 13 2019: (Start)

Conjecture for rightmost terms: A122722(n) = n! * T(n, n*(n+1)/2) for n >= 1.

Conjectures for column k >= 0: Sum_{0 <= s <= 2*k + 1} (-1)^s * binomial(2*k+1, s) * T(n-s, k) = 0 for n >= 2*k+2.

Sum_{0 <= s <= 2*k} (-1)^s * binomial(2*k, s) * T(n-s, k) = A011781(k) for n >= 2*k+1. (End)

Example

Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
  1;
  1,  3;
  1,  9,  24,   18;
  1, 18, 126,  434,   762,   630,   187;
  1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;
  ...

Crossrefs

Cf. A045943 (column k=1), A011555, A011556, A011781, A122722, A326367 (k=2), A326368 (k=3), A326369 (k=4), A000124 (row lengths).

Sequence in context: A160568 A157403 A225118 * A105951 A038202 A128415

Adjacent sequences: A273461 A273462 A273463 * A273465 A273466 A273467

Keyword

tabf,nonn

Author

R. J. Mathar, May 23 2016

Status

approved