A curious sum

[franktaw at netscape.net]

Consider first the sequence which counts up to k (A002260):

1, 1,2, 1,2,3, 1,2,3,4, 1,2,3,4,5, 1,2,3,4,5,6, ...

Now, alternate the signs in  each group so that the final sign is 
positive:

1, -1,2, 1,-2,3, -1,2,-3,4, 1,-2,3,-4,5, -1,2,-3,4,-5,6 ...

Take the cumulative sum of this sequence:

1, 0,2, 3,1,4, 3,5,2,6, 7,5,8,4,9, 8,10,7,11,6,12, ...

It appears that every positive integer occurs exactly twice in this 
sequence (with 0 occurring only once).

(Note that the last elements of each group are the quarter-squares 
(A002620), and the next to last elements are the same sequence, shifted 
left by two.  These next to last elements are the minimum in the group.)

Franklin T. Adams-Watters