A001197 Zarankiewicz's problem k_2(n). (Formerly M3300 N1330)

4, 7, 10, 13, 17, 22, 25, 30, 35, 40, 46, 53, 57, 62, 68, 75, 82, 89, 97, 106, 109, 116, 123

Offset

2,1

Comments

a(n) is the minimum number k_2(n) such that any n X n matrix having that number of nonzero entries has a 2 X 2 submatrix with only nonzero entries. - M. F. Hasler, Sep 28 2021

a(n) <= (1 + sqrt(4*n-3))*n/2 + 1. - Max Alekseyev, Apr 03 2022

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 291.

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.

Richard J. Nowakowski, Zarankiewicz's Problem, PhD Dissertation, University of Calgary, 1978, page 202.

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

Table of n, a(n) for n=2..24.

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

R. K. Guy, A many-facetted problem of Zarankiewicz, Lect. Notes Math. 110 (1969), 129-148.

Formula

a(n) = A072567(n) + 1. - Rob Pratt, Aug 09 2019

a(n) = n^2 - A347472(n) = n^2 - A350296(n) + 1. - Andrew Howroyd, Dec 26 2021

Crossrefs

Cf. A001198 (k_3), A072567, A339635, A347472, A350296.

Cf. also A006613 - A006626 (other sizes, in particular A006616 = k_4).

Sequence in context: A143455 A310684 A087065 * A276874 A310685 A310686

Adjacent sequences: A001194 A001195 A001196 * A001198 A001199 A001200

Keyword

nonn,hard,more

Author

N. J. A. Sloane

Extensions

Nowakowski's thesis, directed by Guy, corrected Guy's value for a(15) and supplied a(16)-a(21) entered by Don Knuth, Aug 13 2014

a(1) deleted following a suggestion from M. F. Hasler. - N. J. A. Sloane, Oct 22 2021

a(22)-a(24) from Jeremy Tan, Jan 23 2022

Status

approved