OFFSET
1,1
COMMENTS
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 initial terms and the corresponding compositions:
9: (3,1) 183: (2,1,2,1,1,1)
34: (4,2) 189: (2,1,1,1,2,1)
39: (3,1,1,1) 194: (1,5,2)
45: (2,1,2,1) 199: (1,4,1,1,1)
49: (1,4,1) 205: (1,3,1,2,1)
57: (1,1,3,1) 209: (1,2,4,1)
132: (5,3) 217: (1,2,1,3,1)
139: (4,2,1,1) 226: (1,1,4,2)
142: (4,1,1,2) 231: (1,1,3,1,1,1)
149: (3,2,2,1) 237: (1,1,2,1,2,1)
154: (3,1,2,2) 241: (1,1,1,4,1)
159: (3,1,1,1,1,1) 249: (1,1,1,1,3,1)
161: (2,5,1) 520: (6,4)
169: (2,2,3,1) 531: (5,3,1,1)
178: (2,1,3,2) 534: (5,2,1,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Select[Range[0, 100], sats[stc[#]]==-2&]
CROSSREFS
These compositions are counted by A088218.
These are the positions of 2's in A344618.
The case of partitions of 2n is A344741.
The opposite (negative 2) version is A345923.
The version for unreversed alternating sum is A345925.
The version for Heinz numbers of partitions is A345961.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A120452 counts partitions of 2n with reverse-alternating sum 2.
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, Jul 10 2021
STATUS
approved