A001843 The coding-theoretic function A(n,4,4). (Formerly M2644 N1052)

1, 1, 3, 7, 14, 18, 30, 35, 51, 65, 91, 105, 140, 157, 198, 228, 285, 315, 385, 419, 498, 550, 650, 702, 819, 877, 1005, 1085, 1240, 1320, 1496, 1583, 1773, 1887, 2109, 2223, 2470, 2593, 2856, 3010, 3311, 3465, 3795, 3959, 4308, 4508, 4900, 5100, 5525, 5737

Offset

4,3

Comments

Maximal number of 4-subsets of an n-set such that any two subsets meet in at most 2 points.

References

CRC Handbook of Combinatorial Designs, 1996, p. 411.

R. K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Georg Fischer, Table of n, a(n) for n = 4..1003

Jingjun Bao and Lijun Ji, The completion determination of optimal (3,4)-packings, Des. Codes Cryptogr. 77, 217-229 (2015); arXiv:1401.2022 [math.CO], 2014.

A. E. Brouwer, Bounds for constant weight binary codes

A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, New table of constant weight codes, IEEE Trans. Info. Theory 36 (1990), 1334-1380.

R. K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]

L. Ji, Asymptotic Determination of the Last Packing Number of Quadruples, Designs, Codes and Cryptography 38 (2006) 83-95.

Index entries for sequences related to A(n,d,w)

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,1,-1,-1,1,0,0,1,-1,-1,1,0,0,-1,1,1,-1).

Formula

See Theorem 1.2 of Bao and Ji, 2015 (Theorem 4.9 in the arXiv preprint, but note the missing parentheses for J(n,4,4) on page 1).

a(n)= +a(n-1) +a(n-2) -a(n-3) +a(n-6) -a(n-7) -a(n-8) +a(n-9) +a(n-12) -a(n-13) -a(n-14) +a(n-15) -a(n-18) +a(n-19) +a(n-20) -a(n-21). - R. J. Mathar, Oct 01 2021

Example

For n=7 use all seven cyclic shifts of 1110100.

Maple

A001843 :=  proc(n)
    floor((n-1)/3* floor((n-2)/2) ) ;
    if modp(n, 6) = 0 then
        floor(n*(%-1)/4) ;
    else
        floor(n*%/4) ;
    end if;
end proc:
seq(A001843(n), n=4..80) ; # R. J. Mathar, Oct 01 2021

Prog

(Python)
[((n-2)//2*(n-1)//3 - int(n%6 == 0)) * n // 4 for n in range(4, 50)]
# Andrey Zabolotskiy, Jan 28 2021

Crossrefs

Sequence in context: A310268 A190700 A267448 * A310269 A033808 A310270

Adjacent sequences: A001840 A001841 A001842 * A001844 A001845 A001846

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Revised by N. J. A. Sloane and Andries E. Brouwer, May 08 2010

Terms a(23) and beyond added, entry edited by Andrey Zabolotskiy, Jan 28 2021

Status

approved