Well, a straightforward translation would be:
var m = Enumerable.Range(1, A.Count - 1)
.Select(i => A[i] + A[i - 1])
.ToList();
But also consider:
var m = A.Skip(1)
.Zip(A, (curr, prev) => curr + prev)
.ToList();
Or using Jon Skeet's extension here:
var m = A.SelectWithPrevious((prev, curr) => prev + curr)
.ToList();
But as Jason Evans points out in a comment, this doesn't help all that much with readability or brevity, considering your existing code is perfectly understandable (and short) and you want to materialize all of the results into a list anyway.
There's nothing really wrong with:
var sumsOfConsecutives = new List<int>();
for(int i = 1; i < A.Count; i++)
sumsOfConsecutives.Add(A[i] + A[i - 1]);
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