A030505 Graham-Sloane-type lower bound on the size of a ternary (n,3,5) constant-weight code.

3, 15, 45, 106, 213, 384, 643, 1014, 1526, 2210, 3100, 4236, 5658, 7411, 9541, 12101, 15144, 18727, 22910, 27758, 33337, 39717, 46971, 55175, 64410, 74758, 86304, 99139, 113355, 129047, 146314, 165257, 185983, 208600, 233219, 259954, 288925, 320252, 354060, 390475

Offset

5,1

Links

Table of n, a(n) for n=5..44.

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 (4,-6,4,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-4,6,-4,1).

Formula

a(n) = ceiling(binomial(n, w) * 2^w / (2*n + 1)) with w = 5.

From Chai Wah Wu, May 12 2026: (Start)

If n is in {1,2,3,4,5,10,31,52}, then a(n) = binomial(n, 5) * 32 / (2*n + 1). Otherwise, a(n) = 1 + floor(binomial(n, 5) * 32 / (2*n + 1)).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-20) - 4*a(n-21) + 6*a(n-22) - 4*a(n-23) + a(n-24) for n > 76.

G.f.: x^5*(-x^71 + 4*x^70 - 6*x^69 + 4*x^68 - x^67 - x^50 + 4*x^49 - 6*x^48 + 4*x^47 - x^46 - x^42 + 4*x^41 - 6*x^40 + 4*x^39 - 2*x^38 + 4*x^37 - 6*x^36 + 4*x^35 - 2*x^34 + 4*x^33 - 6*x^32 + 4*x^31 - x^30 - x^29 + 4*x^28 - 6*x^27 + 3*x^26 + 2*x^25 - 3*x^24 + x^23 - x^21 + 2*x^20 - 5*x^19 - 3*x^18 - 4*x^17 - 7*x^15 - x^14 - 5*x^13 - 6*x^11 - 3*x^10 - 2*x^9 - 5*x^8 - 6*x^6 - 3*x^5 - 2*x^4 - 4*x^3 - 3*x^2 - 3*x - 3)/(x^24 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - x^4 + 4*x^3 - 6*x^2 + 4*x - 1). (End)

Mathematica

a[n_] := Ceiling[Binomial[n, 5] * 2^5 / (2*n + 1)]; Array[a, 40, 5] (* Amiram Eldar, Oct 31 2025 *)

Prog

(Python)
from math import comb
def A030505(n):
    a, b = divmod(comb(n, 5)<<5, (n<<1|1))
    return a+bool(b) # Chai Wah Wu, May 12 2026

Crossrefs

Column k=5 of A390161.

Sequence in context: A094191 A050534 A048099 * A301632 A074355 A201868

Adjacent sequences: A030502 A030503 A030504 * A030506 A030507 A030508

Keyword

nonn,easy

Author

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

Status

approved