OFFSET
0,3
COMMENTS
We define the skew-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 358-th composition is (2,1,3,1,2) so a(358) = 2 - 1 - 3 + 1 + 2 = 1.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[skats[stc[n]], {n, 0, 100}]
CROSSREFS
See link for sequences related to standard compositions.
Positions of positive firsts appear to be A029744.
The reverse version is A357624.
The version for prime indices is A357630.
The version for Heinz numbers of partitions is A357634.
KEYWORD
sign
AUTHOR
Gus Wiseman, Oct 08 2022
STATUS
approved