OFFSET
1,3
COMMENTS
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
The sequence of terms together with the corresponding compositions begins:
0: () 17: (4,1) 37: (3,2,1)
1: (1) 18: (3,2) 38: (3,1,2)
2: (2) 19: (3,1,1) 39: (3,1,1,1)
3: (1,1) 21: (2,2,1) 41: (2,3,1)
4: (3) 22: (2,1,2) 42: (2,2,2)
5: (2,1) 23: (2,1,1,1) 43: (2,2,1,1)
7: (1,1,1) 26: (1,2,2) 44: (2,1,3)
8: (4) 28: (1,1,3) 45: (2,1,2,1)
9: (3,1) 29: (1,1,2,1) 46: (2,1,1,2)
10: (2,2) 31: (1,1,1,1,1) 47: (2,1,1,1,1)
11: (2,1,1) 32: (6) 50: (1,3,2)
13: (1,2,1) 33: (5,1) 52: (1,2,3)
14: (1,1,2) 34: (4,2) 53: (1,2,2,1)
15: (1,1,1,1) 35: (4,1,1) 55: (1,2,1,1,1)
16: (5) 36: (3,3) 56: (1,1,4)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Select[Range[0, 100], ats[stc[#]]>=0&]
CROSSREFS
These compositions are counted by A116406.
These are the positions of terms >= 0 in A124754.
The version for prime indices is A344609.
The reverse-alternating sum version is A345914.
The opposite (k <= 0) version is A345915.
The strict (k > 0) version is A345917.
The complement is A345919.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A345197 counts compositions by sum, length, and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 04 2021
STATUS
approved