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A263542 Triangle T(M, N): Number of M X N matrices where 1<N<=M, all elements are distinct, all elements are at least 0 and at most M*N-1, and every 2 X 2 block of elements has the same sum. 0
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

Table of n, a(n) for n=2..16.

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.)