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A326369 Number of tilings of an equilateral triangle of side length n with unit triangles (of side length 1) and exactly four unit "lozenges" or "diamonds" (also of side length 1). 7
0, 0, 0, 762, 12699, 90270, 417435, 1478160, 4354497, 11203269, 25970895, 55414395, 110505120, 208300257, 374375664, 645922095, 1075615380, 1736379630, 2727171042, 4179918384, 6267764745, 9214763640, 13307191065, 18906643602, 26465101179, 36542141595, 49824502425 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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 (9,-36,84,-126,126,-84,36,-9,1).

FORMULA

a(n) = (3/128)*(n-3)*(n-2)*(9*n^6 + 9*n^5 - 135*n^4 - 81*n^3 + 670*n^2 + 104*n - 1216) for n >= 2 (proved by Greg Dresden and Eldin Sijaric).

From Colin Barker, Jul 01 2019: (Start)

G.f.: 3*x^4*(254 + 1947*x + 1137*x^2 - 613*x^3 + 87*x^4 + 33*x^5 - 10*x^6) / (1 - x)^9.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>10.

a(n) = (3/128)*(-7296 + 6704*n + 2284*n^2 - 3732*n^3 + 265*n^4 + 648*n^5 - 126*n^6 - 36*n^7 + 9*n^8) for n>1.

(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=4, here is one of the 762 different tiling of the triangle of side length 4 with exactly four lozenges:

            o

           / \

          o - o

         / \ / \

        o - o   o

       /   / \ / \

      o - o - o - o

     /   / \ / \   \

    o - o - o - o - o

MATHEMATICA

Rest@ CoefficientList[Series[3 x^4*(254 + 1947 x + 1137 x^2 - 613 x^3 + 87 x^4 + 33 x^5 - 10 x^6)/(1 - x)^9, {x, 0, 27}], x] (* Michael De Vlieger, Jul 07 2019 *)

PROG

(PARI) concat([0, 0, 0], Vec(3*x^4*(254 + 1947*x + 1137*x^2 - 613*x^3 + 87*x^4 + 33*x^5 - 10*x^6) / (1 - x)^9 + O(x^40))) \\ Colin Barker, Jul 01 2019

CROSSREFS

Cf. A326367, A326368. Column 4 of A273464.

Sequence in context: A253615 A253616 A253611 * A210081 A083645 A145718

Adjacent sequences:  A326366 A326367 A326368 * A326370 A326371 A326372

KEYWORD

nonn,easy

AUTHOR

Greg Dresden, Jul 01 2019

STATUS

approved

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Last modified January 25 23:03 EST 2020. Contains 331270 sequences. (Running on oeis4.)