%I M2644 N1052 #43 Sep 25 2022 07:19:28
%S 1,1,3,7,14,18,30,35,51,65,91,105,140,157,198,228,285,315,385,419,498,
%T 550,650,702,819,877,1005,1085,1240,1320,1496,1583,1773,1887,2109,
%U 2223,2470,2593,2856,3010,3311,3465,3795,3959,4308,4508,4900,5100,5525,5737
%N The coding-theoretic function A(n,4,4).
%C Maximal number of 4-subsets of an n-set such that any two subsets meet in at most 2 points.
%D CRC Handbook of Combinatorial Designs, 1996, p. 411.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Georg Fischer, <a href="/A001843/b001843.txt">Table of n, a(n) for n = 4..1003</a>
%H Jingjun Bao and Lijun Ji, <a href="https://doi.org/10.1007/s10623-014-0001-2">The completion determination of optimal (3,4)-packings</a>, Des. Codes Cryptogr. 77, 217-229 (2015); arXiv:<a href="https://arxiv.org/abs/1401.2022">1401.2022</a> [math.CO], 2014.
%H A. E. Brouwer, <a href="http://www.win.tue.nl/~aeb/codes/Andw.html">Bounds for constant weight binary codes</a>
%H A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, <a href="http://dx.doi.org/10.1109/18.59932">New table of constant weight codes</a>, IEEE Trans. Info. Theory 36 (1990), 1334-1380.
%H R. K. Guy, <a href="/A001197/a001197.pdf">A problem of Zarankiewicz</a>, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
%H L. Ji, <a href="http://dx.doi.org/10.1007/s10623-004-5662-9">Asymptotic Determination of the Last Packing Number of Quadruples</a>, Designs, Codes and Cryptography 38 (2006) 83-95.
%H <a href="/index/Aa#Andw">Index entries for sequences related to A(n,d,w)</a>
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,0,1,-1,-1,1,0,0,1,-1,-1,1,0,0,-1,1,1,-1).
%F 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).
%F 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
%e For n=7 use all seven cyclic shifts of 1110100.
%p A001843 := proc(n)
%p floor((n-1)/3* floor((n-2)/2) ) ;
%p if modp(n,6) = 0 then
%p floor(n*(%-1)/4) ;
%p else
%p floor(n*%/4) ;
%p end if;
%p end proc:
%p seq(A001843(n),n=4..80) ; # _R. J. Mathar_, Oct 01 2021
%o (Python)
%o [((n-2)//2*(n-1)//3 - int(n%6 == 0)) * n // 4 for n in range(4, 50)]
%o # _Andrey Zabolotskiy_, Jan 28 2021
%K nonn,nice,easy
%O 4,3
%A _N. J. A. Sloane_
%E Revised by _N. J. A. Sloane_ and _Andries E. Brouwer_, May 08 2010
%E Terms a(23) and beyond added, entry edited by _Andrey Zabolotskiy_, Jan 28 2021