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

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

Offset: 3

Author

Mattias Svanstrom (mattias(AT)isy.liu.se)

Keywords

• nonn
• easy

Comments

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

Links

• Craig Knecht, Maximum number of enneiamonds in a hexagon.
• Katherine E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051 [math.NT], 2011-2014.
• M. Svanstrom, A lower bound for ternary constant weight codes, IEEE Trans. on Information Theory, Vol. 43, No. 5 (Sep. 1997), pp. 1630-1632.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

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)).

Programs

Mathematica

LinearRecurrence[{2,-1,1,-2,1},{2,6,10,16,24},50] (* Harvey P. Dale, Mar 03 2016 *)

Python

def A030511(n): return ((n-1)**2<<1)//3 # Chai Wah Wu, Aug 04 2025

Formula

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*A000212(n-1). - R. J. Mathar, Aug 25 2011

a(n) = floor( (2/3)*(n-1)^2 ). - Wesley Ivan Hurt, Jun 19 2013

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