A326368 Number of tilings of an equilateral triangle of side length n with unit triangles (of side length 1) and exactly three unit "lozenges" or "diamonds" (also of side length 1).

0, 0, 18, 434, 2814, 11127, 33365, 83568, 184254, 369254, 686952, 1203930, 2009018, 3217749, 4977219, 7471352, 10926570, 15617868, 21875294, 30090834, 40725702, 54318035, 71490993, 92961264, 119547974, 152182002, 191915700, 239933018, 297560034, 366275889

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 (7,-21,35,-35,21,-7,1).

Formula

a(n) = (1/16)*(n-2)*(9*n^5 - 9*n^4 - 81*n^3 + 81*n^2 + 160*n - 192) for n >= 2 (proved by Greg Dresden and E. Sijaric).

From Colin Barker, Jul 02 2019: (Start)

G.f.: x^3*(18 + 308*x + 154*x^2 - 87*x^3 + 10*x^4 + 2*x^5) / (1 - x)^7.

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>8.

(End)

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 18 different tiling of the triangle of side length 3 with exactly three lozenges:
          o
         / \
        o   o
       / \ / \
      o - o   o
     /   / \ / \
    o - o - o - o

Mathematica

Rest@ CoefficientList[Series[x^3*(18 + 308 x + 154 x^2 - 87 x^3 + 10 x^4 + 2 x^5)/(1 - x)^7, {x, 0, 30}], x] (* Michael De Vlieger, Jul 07 2019 *)

Prog

(PARI) concat([0, 0], Vec(x^3*(18 + 308*x + 154*x^2 - 87*x^3 + 10*x^4 + 2*x^5) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Jul 02 2019

Crossrefs

Cf. A273464, A326367, A326369.

Sequence in context: A172135 A005477 A361549 * A197343 A289941 A254211

Adjacent sequences: A326365 A326366 A326367 * A326369 A326370 A326371

Keyword

nonn,easy

Author

Greg Dresden, Jul 01 2019

Status

approved