[seqfan] distinct sums in a square

[hv at crypt.org]

Define a(n) as the least k such that an n x n grid of non-negative
integers summing to k can be found, in which each element when added
to its orthogonal neighbours yields a distinct sum.

I believe the sequence starts 0, 6, 9, 27 which is not in the OEIS.

I have (not necessarily minimal) candidates for a(5) and a(6), and no
candidate for n > 6. I'd appreciate confirmation of at least a(1) .. a(4)
before I submit to the OEIS - I have at best medium confidence in the
correctness of the code I've written to get these values.

For n > 4, I believe [1] we need a(n) >= (n^4 - n^2 + 14) / 10, giving
a(5) > 61.4 and a(6) > 127.4, so my candidates approach but do not hit
the theoretical optimum. The only upper bound I have is the relatively
useless a(n) <= 2^(n^2) - 1.

Examples (need fixed font!):

a(1) = 0:                 element sums
  0                       0

a(2) = 6:
  0 1                     3 4
  2 3                     5 6

a(3) = 9:
  0 0 0                   0 1 3
  0 1 3                   2 8 4
  1 4 0                   5 6 7

a(4) = 27:
  0 0 0 0                  0  1  3  7
  0 1 3 7                  4 10 14 11
  3 6 3 1                  9 15 13 12
  0 2 0 1                 11  8  6  2

a(5) <= 63:
  0  0  0  0  0            0  9  4 13  1
  0  9  4 13  1           10 19 26 22 14
  1  6  0  4  0            7 21 17 25  5
  0  5  3  8  0            6 16 20 18  8
  0  2  4  3  0            2 11 12 15  3

a(6) <= 134: 
  0  0  0  0  0  0         0  4  6  7 12  1
  0  4  6  7 12  1         5 23 21 26 29 13
  1 13  4  1  9  0        14 30 24 22 33 10
  0  8  0  1 11  0         9 37 19 25 38 11
  0 16  6 12 17  0        16 32 34 36 43 17
  0  2  0  0  3  0         2 18  8 15 20  3


Ideas, constructions, better constraints or more values all welcome.

Hugo

---

[1] Justification of lower bound: letting S_c be the sum of the corner
elements and S_e the sum of the non-corner edge elements, then the total
of all the element sums is 5a(n) - S_e - 2S_c. However they are required
to be n^2 distinct non-negative integers, which must sum to at least
T(n^2 - 1) = n^2 (n^2 - 1) / 2.

For n > 4, making the element sums at the corners distinct requires edge
or corner elements contributing to sums of at least { 0, 1, 2, 3 }.
The edge pieces adjacent to the corner with 0 sum must have their other
adjacent edge differ by at least 1, so we have S_e + 2S_c >= 7.

Thus a(n) >= (T(n^2 - 1) + 7) / 5 = (n^4 - n^2 + 14) / 10.