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

Offset: 3

Author

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

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 3..10000
• M. Svanstrom, A lower bound for ternary constant weight codes, IEEE Trans. on Information Theory, Vol. 43, pp. 1630-1632, Sep. 1997.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Programs

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

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)