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A326369
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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).
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7
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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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).
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)
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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