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%I #30 Sep 18 2019 04:58:47
%S 0,0,18,434,2814,11127,33365,83568,184254,369254,686952,1203930,
%T 2009018,3217749,4977219,7471352,10926570,15617868,21875294,30090834,
%U 40725702,54318035,71490993,92961264,119547974,152182002,191915700,239933018,297560034,366275889
%N 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).
%H Colin Barker, <a href="/A326368/b326368.txt">Table of n, a(n) for n = 1..1000</a>
%H Richard J. Mathar, <a href="https://arxiv.org/abs/1909.06336">Lozenge tilings of the equilateral triangle</a>, arXiv:1909.06336 [math.CO], 2019.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F 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).
%F From _Colin Barker_, Jul 02 2019: (Start)
%F G.f.: x^3*(18 + 308*x + 154*x^2 - 87*x^3 + 10*x^4 + 2*x^5) / (1 - x)^7.
%F 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.
%F (End)
%e We can represent a unit triangle this way:
%e o
%e / \
%e o - o
%e and a unit "lozenge" or "diamond" has these three orientations:
%e o
%e / \ o - o o - o
%e o o and / / and also \ \
%e \ / o - o o - o
%e o
%e and for n=3, here is one of the 18 different tiling of the triangle of side length 3 with exactly three lozenges:
%e o
%e / \
%e o o
%e / \ / \
%e o - o o
%e / / \ / \
%e o - o - o - o
%t 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 *)
%o (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
%Y Cf. A273464, A326367, A326369.
%K nonn,easy
%O 1,3
%A _Greg Dresden_, Jul 01 2019