cp's OEIS Frontend

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).

%I A326367 #50 Sep 18 2019 04:58:40
%S A326367 0,0,24,126,387,915,1845,3339,5586,8802,13230,19140,26829,36621,48867,
%T A326367 63945,82260,104244,130356,161082,196935,238455,286209,340791,402822,
%U A326367 472950,551850,640224,738801,848337,969615,1103445,1250664,1412136,1588752,1781430,1991115
%N 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).
%H A326367 Colin Barker, <a href="/A326367/b326367.txt">Table of n, a(n) for n = 1..1000</a>
%H A326367 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 A326367 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A326367 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 A326367 From _Colin Barker_, Jul 01 2019: (Start)
%F A326367 G.f.: 3*x^3*(4 - x)*(2 + x) / (1 - x)^5.
%F A326367 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 A326367 (End)
%F A326367 E.g.f.: (3/8)*exp(x)*x^2*(32 + 24*x + 3*x^2). - _Stefano Spezia_, Jul 01 2019
%e A326367 We can represent a unit triangle this way:
%e A326367        o
%e A326367       / \
%e A326367      o - o
%e A326367 and a unit "lozenge" or "diamond" has these three orientations:
%e A326367      o
%e A326367     / \          o - o            o - o
%e A326367    o   o  and   /   /   and also   \   \
%e A326367     \ /        o - o                o - o
%e A326367      o
%e A326367 and for n=3, here is one of the 24 different tiling of the triangle of side length 3 with exactly two lozenges:
%e A326367           o
%e A326367          / \
%e A326367         o   o
%e A326367        / \ / \
%e A326367       o - o - o
%e A326367      /   / \ / \
%e A326367     o - o - o - o
%t A326367 Rest@ CoefficientList[Series[3 x^3*(4 - x) (2 + x)/(1 - x)^5, {x, 0, 37}], x] (* _Michael De Vlieger_, Jul 04 2019 *)
%o A326367 (PARI) concat([0,0], Vec(3*x^3*(4 - x)*(2 + x) / (1 - x)^5 + O(x^40))) \\ _Colin Barker_, Jul 01 2019
%Y A326367 Cf. A273464, A326368, A326369.
%K A326367 nonn,easy
%O A326367 1,3
%A A326367 _Greg Dresden_, Jul 01 2019