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 A263542 Triangle T(M, N): Number of M X N matrices where 1
 24, 112, 376, 768, 2160, 20352, 5376, 5904, 86208, 51840, 64512, 56736, 1628352, 1342656, 44084736 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS This sequence is given in this order: (2,2), (3,2), (3,3), (4,2), (4,3), (4,4), etc. The idea of the program below is that the first row, first column, and the (1,1)th element uniquely determine the rest of the matrix. Hence, all permutations of m+n integers in the range 0..m*n-1 are generated to fill the first row, first column, and (1,1). Then the empty spots in the matrix are filled in and if at any point a condition is violated (duplicate, < 0, >= M*N), the program immediately moves on to the next permutation. Much of the conversation in the main chat room of the Programming Puzzles and Code Golf Stack Exchange site - the Nineteenth Byte - following the linked message in the Links section deals with finding the terms of this sequence. Observation: at least the first 15 terms are divisible by 8. - Omar E. Pol, Oct 20 2015, Nov 21 2015 When M and N are both even, the block sum is 2(MN-1). When one or both is odd the block sum can vary: e.g., for M=N=3, it varies from 12 to 20. - Peter J. Taylor, Oct 21 2015 When M and N are both even, all solutions are toroidal: the block sums created by wrapping from the last column to the first column or the last row to the first row also equal 2(MN-1). When one of M or N is even, all solutions are cylindrical, with wrapping in the even dimension, but they are toroidal only in the trivial case of Odd X 2. When both M and N are odd, except in the trivial case of 1 X 1, solutions do not wrap in either direction. - Peter J. Taylor, Oct 21 2015 LINKS The Nineteenth Byte, Originating chat message, ChatRoom. EXAMPLE One 3 X 3 solution (with a sum of 19) is:    0 4 2    8 7 6    3 1 5 One 4 X 4 solution (with a sum of 30) is:     0  3  4  7    12 15  8 11     1  2  5  6    13 14  9 10 One 5 X 5 solution (with a sum of 48) is:     0 24  1 23  2     9 15  8 16  7    10 14 11 13 12    19  5 18  6 17    20  4 21  3 22 The triangle T(M, N) begins: M\N    2      3       4       5        6 ... 2:    24 3:   112    376 4:   768   2160   20352 5:  5376   5904   86208   51840 6: 64512  56736 1628352 1342656 44084736 ...reformatted. - Wolfdieter Lang, Dec 16 2015 PROG (Python 3) from itertools import permutations as P n = 4; m = 4; permutes = P(range(m*n), m+n); counter = 0 for p in permutes:   grid = [p[:n]]   for i in range(m-1):     grid.append([p[n+i]]+[-1]*(n-1))   grid[1][1] = p[-1]   s = p[0]+p[1]+p[n]+p[-1]   has = list(p)   fail = 0   for y in range(1, m):     for x in range(1, n):       if x == y == 1: continue       r = s - (grid[y-1][x-1] + grid[y-1][x] + grid[y][x-1])       if r not in has and 0 <= r < m*n:         grid[y][x]=r         has.append(r)       else:        fail = 1        break     if fail: break   if not fail:     counter += 1 print(counter) CROSSREFS Sequence in context: A103473 A162451 A307859 * A281133 A064595 A064591 Adjacent sequences:  A263539 A263540 A263541 * A263543 A263544 A263545 KEYWORD nonn,tabl,more AUTHOR Lee Burnette, Oct 20 2015 STATUS approved

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)