A030503 Graham-Sloane-type lower bound on the size of a ternary (n,3,3) constant-weight code.

2, 4, 8, 13, 19, 27, 36, 46, 58, 71, 85, 101, 118, 136, 156, 177, 199, 223, 248, 274, 302, 331, 361, 393, 426, 460, 496, 533, 571, 611, 652, 694, 738, 783, 829, 877, 926, 976, 1028, 1081, 1135, 1191, 1248, 1306, 1366, 1427, 1489, 1553, 1618, 1684, 1752, 1821

Offset

3,1

Links

Robert Israel, Table of n, a(n) for n = 3..10000

Mattias Svanstrom, A lower bound for ternary constant weight codes, IEEE Trans. on Information Theory, Vol. 43 (Sep. 1997), pp. 1630-1632.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

a(n) = ceiling(binomial(n, w) * 2^w / (2*n + 1)) with w = 3.

Conjectures from Colin Barker, Aug 02 2019: (Start)

G.f.: x^3*(2 + 2*x^2 - x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)).

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 7. (End)

From Robert Israel, Jul 09 2020: (Start)

Conjectures confirmed.

a(n) = (2*n^2-7*n+8)/3 if n == 1 (mod 3), otherwise a(n) = (2*n^2-7*n+9)/3. (End)

E.g.f.: (2*exp(x)*(26 - 15*x + 6*x^2) - 9*(6 + 2*x + x^2)+ 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, May 12 2026

For n>2, a(n) = 1+floor(8*binomial(n,3)/(2*n+1)). - Chai Wah Wu, May 12 2026

Maple

g:= n -> (2*n^2-7*n+`if`(n mod 3 = 1, 8, 9))/3:
map(g, [$3..100]); # Robert Israel, Jul 09 2020

Mathematica

a[n_] := Ceiling[Binomial[n, 3] * 2^3 / (2*n + 1)]; Array[a, 52, 3] (* Amiram Eldar, Oct 31 2025 *)

Prog

(Python)
from math import comb
def A030503(n): return 1+(comb(n, 3)<<3)//(2*n+1) if n>2 else 0 # Chai Wah Wu, May 12 2026

Crossrefs

Cf. A030504, A030505, A030506, A030507, A030508, A030509, A030510.

Sequence in context: A379793 A186752 A360512 * A245094 A164486 A084684

Adjacent sequences: A030500 A030501 A030502 * A030504 A030505 A030506

Keyword

nonn,easy

Author

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

Status

approved