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A030511 Graham-Sloane-type lower bound on the size of a ternary (n,3,3) constant-weight code.

%I A030511 #59 Aug 04 2025 18:39:35
%S A030511 2,6,10,16,24,32,42,54,66,80,96,112,130,150,170,192,216,240,266,294,
%T A030511 322,352,384,416,450,486,522,560,600,640,682,726,770,816,864,912,962,
%U A030511 1014,1066,1120,1176,1232,1290,1350,1410,1472,1536,1600
%N A030511 Graham-Sloane-type lower bound on the size of a ternary (n,3,3) constant-weight code.
%C A030511 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
%C A030511 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
%H A030511 Craig Knecht, <a href="/A030511/a030511.png">Maximum number of enneiamonds in a hexagon</a>.
%H A030511 Katherine E. Stange, <a href="http://arxiv.org/abs/1108.3051">Integral points on elliptic curves and explicit valuations of division polynomials</a> arXiv:1108.3051 [math.NT], 2011-2014.
%H A030511 M. Svanstrom, <a href="https://doi.org/10.1109/18.623164">A lower bound for ternary constant weight codes</a>, IEEE Trans. on Information Theory, Vol. 43, No. 5 (Sep. 1997), pp. 1630-1632.
%H A030511 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A030511 a(n) = 2 * (n - 1)^2 / 3 if n==1 (mod 3), a(n) = 2 * n * (n - 2) / 3 otherwise.
%F A030511 G.f.: -2*x^3*(1 + x) / ( (1 + x + x^2)*(x - 1)^3 ). - _R. J. Mathar_, Aug 25 2011
%F A030511 a(n) = 2*A000212(n-1). - _R. J. Mathar_, Aug 25 2011
%F A030511 a(n) = floor( (2/3)*(n-1)^2 ). - _Wesley Ivan Hurt_, Jun 19 2013
%F A030511 a(n) = (2*(n - 2)*n - (-1)^floor(2*(n-2)/3) + 1)/3. - _Bruno Berselli_, Aug 08 2013
%F A030511 a(n) = a(n-1) + 2*floor((n-1)*2/3). - _Gionata Neri_, Apr 26 2015
%F A030511 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
%F A030511 Sum_{n>=3} 1/a(n) = Pi^2/36 + Pi/(4*sqrt(3)) + 3/8. - _Amiram Eldar_, Sep 24 2022
%F A030511 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
%t A030511 LinearRecurrence[{2,-1,1,-2,1},{2,6,10,16,24},50] (* _Harvey P. Dale_, Mar 03 2016 *)
%o A030511 (Python)
%o A030511 def A030511(n): return ((n-1)**2<<1)//3 # _Chai Wah Wu_, Aug 04 2025
%Y A030511 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)).
%K A030511 nonn,easy
%O A030511 3,1
%A A030511 Mattias Svanstrom (mattias(AT)isy.liu.se)