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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). 7
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 (list; graph; refs; listen; history; text; internal format)
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: A215229 A172135 A005477 * A197343 A289941 A254211

Adjacent sequences:  A326365 A326366 A326367 * A326369 A326370 A326371

KEYWORD

nonn,easy

AUTHOR

Greg Dresden, Jul 01 2019

STATUS

approved

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Last modified September 24 03:42 EDT 2020. Contains 337316 sequences. (Running on oeis4.)