OFFSET
1,2
COMMENTS
We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...
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.
LINKS
EXAMPLE
The sequence together with the corresponding compositions begins:
0: ()
11: (2,1,1)
15: (1,1,1,1)
37: (3,2,1)
38: (3,1,2)
45: (2,1,2,1)
46: (2,1,1,2)
53: (1,2,2,1)
54: (1,2,1,2)
55: (1,2,1,1,1)
59: (1,1,2,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Select[Range[0, 100], halfats[Reverse[stc[#]]]==0&]
CROSSREFS
See link for sequences related to standard compositions.
The alternating form is A344619.
Positions of zeros in A357622.
The non-reverse version is A357625.
The version for prime indices is A357631.
The version for Heinz numbers of partitions is A357635.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 08 2022
STATUS
approved