OFFSET
1,1
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:
6: (1,2)
20: (2,3)
25: (1,3,1)
27: (1,2,1,1)
30: (1,1,1,2)
72: (3,4)
81: (2,4,1)
83: (2,3,1,1)
86: (2,2,1,2)
92: (2,1,1,3)
98: (1,4,2)
101: (1,3,2,1)
103: (1,3,1,1,1)
106: (1,2,2,2)
109: (1,2,1,2,1)
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[#]]==-1&]
CROSSREFS
These compositions are counted by A001791.
A version using runs of binary digits is A031444.
These are the positions of -1's in A124754.
The opposite (positive 1) version is A345909.
The reverse version is A345912.
The version for alternating sum of prime indices is A345959.
A011782 counts compositions.
A097805 counts compositions by sum and 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 01 2021
STATUS
approved