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 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 (list; graph; refs; listen; history; text; internal format)
 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 all-ones 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 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

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Last modified February 2 10:39 EST 2023. Contains 360011 sequences. (Running on oeis4.)