

A347472


Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 2 X 2 zero submatrix.


4



0, 2, 6, 12, 19, 27, 39, 51, 65, 81, 98, 116, 139, 163, 188, 214, 242, 272, 303, 335, 375, 413, 453
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OFFSET

2,2


COMMENTS

Related to Zarankiewicz's problem k_2(n) (cf. A001197 and other crossrefs) which asks the converse: how many 1's must be in an n X n {0,1}matrix in order to guarantee the existence of an allones 2 X 2 submatrix. This complementarity leads to the given formula which was used to compute the given values.
See A347473 and A347474 for the similar problem with a 3 X 3 resp. 4 X 4 zero submatrix.


LINKS

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


FORMULA

a(n) = n^2  A001197(n).
a(n) = A350296(n)  1.  Andrew Howroyd, Dec 23 2021


EXAMPLE

For n = 2, there must not be any nonzero entry in an n X n = 2 X 2 matrix, if one wants a 2 X 2 zero submatrix, whence a(2) = 0.
For n = 3, having at most 2 nonzero entries in the n X n matrix still guarantees that there is a 2 X 2 zero submatrix (delete the row of the first nonzero entry and then the column of the remaining nonzero entry, if any), but if one allows 3 nonzero entries and they are placed on the diagonal, then there is no 2 X 2 zero submatrix. Hence, a(3) = 2.


CROSSREFS

Cf. A001197 (k_2(n)), A001198 (k_3(n)), A006613  A006626.
Cf. A347473, A347474 (analog for 3 X 3 resp. 4 X 4 zero submatrix).
Cf. A350296.
Sequence in context: A104969 A065005 A299016 * A340663 A354759 A139084
Adjacent sequences: A347469 A347470 A347471 * A347473 A347474 A347475


KEYWORD

nonn,hard,more


AUTHOR

M. F. Hasler, Sep 28 2021


EXTENSIONS

a(22)a(24) computed from A001197 by Max Alekseyev, Feb 08 2022


STATUS

approved



