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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). 7

%I

%S 0,0,24,126,387,915,1845,3339,5586,8802,13230,19140,26829,36621,48867,

%T 63945,82260,104244,130356,161082,196935,238455,286209,340791,402822,

%U 472950,551850,640224,738801,848337,969615,1103445,1250664,1412136,1588752,1781430,1991115

%N 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).

%H Colin Barker, <a href="/A326367/b326367.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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F 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).

%F From _Colin Barker_, Jul 01 2019: (Start)

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

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

%F (End)

%F E.g.f.: (3/8)*exp(x)*x^2*(32 + 24*x + 3*x^2). - _Stefano Spezia_, Jul 01 2019

%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 24 different tiling of the triangle of side length 3 with exactly two lozenges:

%e o

%e / \

%e o o

%e / \ / \

%e o - o - o

%e / / \ / \

%e o - o - o - o

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

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

%Y Cf. A273464, A326368, A326369.

%K nonn,easy

%O 1,3

%A _Greg Dresden_, Jul 01 2019

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Last modified April 6 15:23 EDT 2020. Contains 333276 sequences. (Running on oeis4.)