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A030511
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Graham-Sloane-type lower bound on the size of a ternary (n,3,3) constant-weight code.
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15
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2, 6, 10, 16, 24, 32, 42, 54, 66, 80, 96, 112, 130, 150, 170, 192, 216, 240, 266, 294, 322, 352, 384, 416, 450, 486, 522, 560, 600, 640, 682, 726, 770, 816, 864, 912, 962, 1014, 1066, 1120, 1176, 1232, 1290, 1350, 1410, 1472, 1536, 1600
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OFFSET
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3,1
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COMMENTS
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With a different offset this is the elliptic troublemaker sequence R_n(2,6) (also sequence R_n(4,6)) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 12 2013
a(n) is the maximum number of equilateral triangles that can be formed by adding n+1 straight lines on an infinite grid of regular hexagons. - Dhairya Baxi, Sep 03 2022
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LINKS
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FORMULA
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a(n) = 2 * (n - 1)^2 / 3 if n==1 (mod 3), a(n) = 2 * n * (n - 2) / 3 otherwise.
G.f.: -2*x^3*(1 + x) / ( (1 + x + x^2)*(x - 1)^3 ). - R. J. Mathar, Aug 25 2011
a(n) = (2*(n - 2)*n - (-1)^floor(2*(n-2)/3) + 1)/3. - Bruno Berselli, Aug 08 2013
a(n) = a(n-1) + 2*floor((n-1)*2/3). - Gionata Neri, Apr 26 2015
a(n) = floor((n-2)*(n-1)/3) + floor((n-1)*n/3) = floor((n-1)*(n+1)/3) + floor((n-1)*(n-3)/3). - Bruno Berselli, Mar 02 2017
Sum_{n>=3} 1/a(n) = Pi^2/36 + Pi/(4*sqrt(3)) + 3/8. - Amiram Eldar, Sep 24 2022
E.g.f.: 2*exp(-x/2)*(exp(3*x/2)*(1 + 3*x*(x - 1)) - cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 28 2022
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MATHEMATICA
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LinearRecurrence[{2, -1, 1, -2, 1}, {2, 6, 10, 16, 24}, 50] (* Harvey P. Dale, Mar 03 2016 *)
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CROSSREFS
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Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), A007590 (= R_n(2,4)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)).
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KEYWORD
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nonn,easy
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AUTHOR
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Mattias Svanstrom (mattias(AT)isy.liu.se)
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STATUS
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approved
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