OFFSET
1,2
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:
1: (1) 26: (1,2,2) 52: (1,2,3)
2: (2) 27: (1,2,1,1) 54: (1,2,1,2)
4: (3) 28: (1,1,3) 56: (1,1,4)
6: (1,2) 30: (1,1,1,2) 59: (1,1,2,1,1)
7: (1,1,1) 31: (1,1,1,1,1) 60: (1,1,1,3)
8: (4) 32: (6) 62: (1,1,1,1,2)
11: (2,1,1) 35: (4,1,1) 64: (7)
12: (1,3) 37: (3,2,1) 67: (5,1,1)
14: (1,1,2) 38: (3,1,2) 69: (4,2,1)
16: (5) 40: (2,4) 70: (4,1,2)
19: (3,1,1) 42: (2,2,2) 72: (3,4)
20: (2,3) 44: (2,1,3) 73: (3,3,1)
21: (2,2,1) 47: (2,1,1,1,1) 74: (3,2,2)
22: (2,1,2) 48: (1,5) 76: (3,1,3)
24: (1,4) 51: (1,3,1,1) 79: (3,1,1,1,1)
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[#]]>0&]
CROSSREFS
The version for prime indices is A000037.
These compositions are counted by A027306.
These are the positions of terms > 0 in A344618.
The weak (k >= 0) version is A345914.
The version for unreversed alternating sum is A345917.
The opposite (k < 0) version is A345920.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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 09 2021
STATUS
approved