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 A376167 #9 Sep 14 2024 06:47:00 %S A376167 4,5,5,6,7,6,7,8,8,7,8,9,10,9,8,9,10,11,11,10,9,10,11,13,13,13,11,10, %T A376167 11,12,14,15,15,14,12,11,12,13,15,16,17,16,15,13,12,13,14,16,18,19,19, %U A376167 18,16,14,13,14,15,17,19,20,22,20,19,17,15,14,15,16,18,21,22,23,23,22,21,18,16,15 %N A376167 Square array read by antidiagonals: T(n,k) = smallest r such that every n X k binary matrix with r ones contains a 2 X 2 submatrix of ones, with n, k >= 2. %D A376167 Donald E. Knuth, The Art of Computer Programming, Vol. 4B: Combinatorial Algorithms, Part 2, Addison-Wesley, 2023, section 7.2.2.2, pp. 289-291. %F A376167 T(n,k) = T(k,n). %F A376167 T(2,k) = k + 2. %F A376167 T(n,2) = n + 2. %e A376167 Array begins (cf. Table 5 in Knuth (2023), section 7.2.2.2, p. 291, where T(n,k) is denoted by Z(m,n)): %e A376167 n\k| 2 3 4 5 6 7 8 9 10 11 12 ... %e A376167 ----------------------------------------------------- %e A376167 2 | 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... %e A376167 3 | 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ... %e A376167 4 | 6, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, ... %e A376167 5 | 7, 9, 11, 13, 15, 16, 18, 19, 21, 22, 23, ... %e A376167 6 | 8, 10, 13, 15, 17, 19, 20, 22, 23, 25, 26, ... %e A376167 7 | 9, 11, 14, 16, 19, 22, 23, 25, 26, 28, 29, ... %e A376167 8 | 10, 12, 15, 18, 20, 23, 25, 27, 29, 31, 33, ... %e A376167 9 | 11, 13, 16, 19, 22, 25, 27, 30, 32, 34, 37, ... %e A376167 10 | 12, 14, 17, 21, 23, 26, 29, 32, 35, 37, 40, ... %e A376167 11 | 13, 15, 18, 22, 25, 28, 31, 34, 37, 40, 43, ... %e A376167 12 | 14, 16, 19, 23, 26, 29, 33, 37, 40, 43, 46, ... %e A376167 ... %e A376167 T(3,4) = 8 because, no matter how 8 ones are arranged in a 3 X 4 matrix, a 2 X 2 submatrix of ones cannot be avoided (in the left configuration below, for example, the submatrix elements are highlighted by parentheses). 7 ones can avoid such submatrix (right). %e A376167 . %e A376167 (1) 0 (1) 1 1 0 1 1 %e A376167 0 1 0 1 0 1 0 1 %e A376167 (1) 1 (1) 0 1 1 0 0 %Y A376167 Cf. A001197 (main diagonal), A347472, A350296. %K A376167 nonn,tabl %O A376167 2,1 %A A376167 _Paolo Xausa_, Sep 13 2024