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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).

%I A326369 #33 Sep 18 2019 04:58:53
%S A326369 0,0,0,762,12699,90270,417435,1478160,4354497,11203269,25970895,
%T A326369 55414395,110505120,208300257,374375664,645922095,1075615380,
%U A326369 1736379630,2727171042,4179918384,6267764745,9214763640,13307191065,18906643602,26465101179,36542141595,49824502425
%N 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).
%H A326369 Colin Barker, <a href="/A326369/b326369.txt">Table of n, a(n) for n = 1..1000</a>
%H A326369 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 A326369 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A326369 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).
%F A326369 From _Colin Barker_, Jul 01 2019: (Start)
%F A326369 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.
%F A326369 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.
%F A326369 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.
%F A326369 (End)
%e A326369 We can represent a unit triangle this way:
%e A326369        o
%e A326369       / \
%e A326369      o - o
%e A326369 and a unit "lozenge" or "diamond" has these three orientations:
%e A326369      o
%e A326369     / \          o - o            o - o
%e A326369    o   o  and   /   /   and also   \   \
%e A326369     \ /        o - o                o - o
%e A326369      o
%e A326369 and for n=4, here is one of the 762 different tiling of the triangle of side length 4 with exactly four lozenges:
%e A326369             o
%e A326369            / \
%e A326369           o - o
%e A326369          / \ / \
%e A326369         o - o   o
%e A326369        /   / \ / \
%e A326369       o - o - o - o
%e A326369      /   / \ / \   \
%e A326369     o - o - o - o - o
%t A326369 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 *)
%o A326369 (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
%Y A326369 Cf. A326367, A326368. Column 4 of A273464.
%K A326369 nonn,easy
%O A326369 1,4
%A A326369 _Greg Dresden_, Jul 01 2019