Here is one solution:
Alower = tril(A, -1);
Aupper = triu(A, 1);
result = Alower(:, 1:end-1) + Aupper(:, 2:end)
Demo:
> A = [1 2 3; 1 2 3; 1 2 3]
A =
1 2 3
1 2 3
1 2 3
> tril(A, -1)(1:end, 1:end-1) + triu(A, 1)(1:end, 2:end)
ans =
2 3
1 3
1 2
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