This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347474 #13 Feb 02 2024 11:01:46
%S A347474 0,2,4,6,12,19,25,34,43,51
%N A347474 Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 4 X 4 zero submatrix.
%C A347474 Related to Zarankiewicz's problem k_4(n) (cf. A006616 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 4 X 4 submatrix. This complementarity leads to the given formula which was used to compute the given values.
%F A347474 a(n) = n^2 - A006616(n).
%F A347474 a(n) = A339635(n,4) - 1. - _Andrew Howroyd_, Dec 23 2021
%e A347474 For n < 4, there is no solution, since there cannot be a 4 X 4 submatrix in a matrix of smaller size.
%e A347474 For n = 4, there must not be any nonzero entry in an n X n = 4 X 4 matrix, if one wants a 4 X 4 zero submatrix, whence a(4) = 0.
%e A347474 For n = 5, having at most 2 nonzero entries in the n X n matrix guarantees that there is a 4 X 4 zero submatrix (delete, e.g., the row with the first nonzero entry, then the column with the second nonzero entry, if any), but if one allows 3 nonzero entries and they are placed on the diagonal, then there is no 4 X 4 zero submatrix. Hence, a(5) = 2.
%Y A347474 Cf. A347472, A347473 (analog for 2 X 2 resp. 3 X 3 zero submatrix).
%Y A347474 Cf. A006616 (k_4(n)), A001198 (k_3(n)), A001197 (k_2(n)), A006613 - A006626.
%Y A347474 Cf. A339635.
%K A347474 nonn,hard,more
%O A347474 4,2
%A A347474 _M. F. Hasler_, Sep 28 2021
%E A347474 a(9)-a(12) from _Andrew Howroyd_, Dec 23 2021
%E A347474 a(13) computed from A006616 by _Max Alekseyev_, Feb 02 2024