A376167 -id:A376167 - cp's OEIS frontend

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

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

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).

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.

Cf. A001198 (k_3), A072567, A339635, A347472, A350296.
Cf. also A006613 - A006626 (other sizes, in particular A006616 = k_4).
Main diagonal of A376167.

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

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

cp's OEIS Frontend

A001197 Zarankiewicz's problem k_2(n).

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