A326367 Number of tilings of an equilateral triangle of side length n with unit triangles (of side length 1) and exactly two unit "lozenges" or "diamonds" (also of side length 1).

0, 0, 24, 126, 387, 915, 1845, 3339, 5586, 8802, 13230, 19140, 26829, 36621, 48867, 63945, 82260, 104244, 130356, 161082, 196935, 238455, 286209, 340791, 402822, 472950, 551850, 640224, 738801, 848337, 969615, 1103445, 1250664, 1412136, 1588752, 1781430, 1991115

Offset

1,3

Links

Colin Barker, Table of n, a(n) for n = 1..1000

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

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (3/8)*(n-2)*(n-1)*(3*n^2 + 3*n - 4) (conjectured by R. J. Mathar, proved by Greg Dresden and E. Sijaric).

From Colin Barker, Jul 01 2019: (Start)

G.f.: 3*x^3*(4 - x)*(2 + x) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

E.g.f.: (3/8)*exp(x)*x^2*(32 + 24*x + 3*x^2). - Stefano Spezia, Jul 01 2019

Example

We can represent a unit triangle this way:
       o
      / \
     o - o
and a unit "lozenge" or "diamond" has these three orientations:
     o
    / \          o - o            o - o
   o   o  and   /   /   and also   \   \
    \ /        o - o                o - o
     o
and for n=3, here is one of the 24 different tiling of the triangle of side length 3 with exactly two lozenges:
          o
         / \
        o   o
       / \ / \
      o - o - o
     /   / \ / \
    o - o - o - o

Mathematica

Rest@ CoefficientList[Series[3 x^3*(4 - x) (2 + x)/(1 - x)^5, {x, 0, 37}], x] (* Michael De Vlieger, Jul 04 2019 *)

Prog

(PARI) concat([0, 0], Vec(3*x^3*(4 - x)*(2 + x) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Jul 01 2019

Crossrefs

Cf. A273464, A326368, A326369.

Sequence in context: A044356 A044737 A305159 * A182186 A188304 A167981

Adjacent sequences: A326364 A326365 A326366 * A326368 A326369 A326370

Keyword

nonn,easy

Author

Greg Dresden, Jul 01 2019

Status

approved