OFFSET
1,2
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: ()
3: (1,1)
6: (1,2)
10: (2,2)
12: (1,3)
13: (1,2,1)
15: (1,1,1,1)
20: (2,3)
24: (1,4)
25: (1,3,1)
27: (1,2,1,1)
30: (1,1,1,2)
36: (3,3)
40: (2,4)
41: (2,3,1)
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 A058622.
These are the positions of terms <= 0 in A124754.
The reverse-alternating version is A345916.
The opposite (k >= 0) version is A345917.
The strictly negative (k < 0) version is A345919.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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 08 2021
STATUS
approved