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 A384616 #43 Jun 30 2025 17:21:00 %S A384616 0,1,8,3,23,58,7,44,115 %N A384616 A(m,n) is the maximum sum of absolute differences of the labels of adjacent vertices of the grid graph P_m X P_n where the m*n labels are exactly 1, 2, ..., m*n. %C A384616 A(m, n) ~ Theta((m*n)^2) (see link). %H A384616 Sela Fried, <a href="/A384616/a384616.pdf">On the maximal sum of absolute differences of certain labelings of the grid graph</a>, 2025. %F A384616 Conjecture: A(m,2) = A067725(m-1) - 1. %e A384616 Array begins (values in parentheses are conjectural): %e A384616 [1] 0 %e A384616 [2] 1 8 %e A384616 [3] 3 23 58 %e A384616 [4] 7 44 115 (216) %e A384616 [5] 11 71 (182) (347) (554) %e A384616 [6] 17 104 (271) (508) (815) (1192) %e A384616 [7] 23 143 (370) (699) (1118) (1639) (2250) %e A384616 [8] 31 (188) (491) (920) (1475) (2156) (2963) (3896) %e A384616 [9] 39 (239) (622) (1171) (1874) (2743) (3766) (4955) (6298) %o A384616 (Python) %o A384616 import itertools %o A384616 import numpy as np %o A384616 def max_difference_sum(m, n): %o A384616 nums = list(range(1, m * n + 1)) %o A384616 max_sum = 0 %o A384616 best_matrix = None %o A384616 for perm in itertools.permutations(nums): %o A384616 matrix = np.array(perm).reshape((m, n)) %o A384616 diff_sum = np.sum(np.abs(matrix[:,1:]-matrix[:,:-1])) + np.sum(np.abs(matrix[1:,:]-matrix[:-1,:])) %o A384616 if diff_sum > max_sum: %o A384616 max_sum = diff_sum %o A384616 best_matrix = matrix.copy() %o A384616 return max_sum, best_matrix %o A384616 for m in range(1, 10): %o A384616 for n in range(1, m+1): %o A384616 max_sum, best = max_difference_sum(m, n) %o A384616 print(max_sum, end=', ') %Y A384616 Column 1 is A047838. %Y A384616 Cf. A067725. %K A384616 nonn,tabl,more %O A384616 1,3 %A A384616 _Sela Fried_, Jun 07 2025