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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

Examples

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

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).

Programs

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

Python

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

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

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