OFFSET
1,2
COMMENTS
The alternating sum of a composition (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:
1: (1) 87: (2,2,1,1,1)
5: (2,1) 90: (2,1,2,2)
7: (1,1,1) 93: (2,1,1,2,1)
18: (3,2) 95: (2,1,1,1,1,1)
21: (2,2,1) 100: (1,3,3)
23: (2,1,1,1) 105: (1,2,3,1)
26: (1,2,2) 107: (1,2,2,1,1)
29: (1,1,2,1) 110: (1,2,1,1,2)
31: (1,1,1,1,1) 114: (1,1,3,2)
68: (4,3) 117: (1,1,2,2,1)
73: (3,3,1) 119: (1,1,2,1,1,1)
75: (3,2,1,1) 122: (1,1,1,2,2)
78: (3,1,1,2) 125: (1,1,1,1,2,1)
82: (2,3,2) 127: (1,1,1,1,1,1,1)
85: (2,2,2,1) 264: (5,4)
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[#]]==1&]
CROSSREFS
The version for prime indices is A001105.
A version using runs of binary digits is A031448.
These are the positions of 1's in A124754.
The opposite (negative 1) version is A345910.
The reverse version is A345911.
The version for Heinz numbers of partitions is A345958.
A011782 counts compositions.
A097805 counts compositions by sum and alternating sum.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 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, Jun 30 2021
STATUS
approved