The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #5 Oct 08 2022 09:39:50
%S 0,1,2,0,3,-1,1,-1,4,-2,0,-2,2,-2,0,0,5,-3,-1,-3,1,-3,-1,1,3,-3,-1,-1,
%T 1,-1,1,1,6,-4,-2,-4,0,-4,-2,2,2,-4,-2,0,0,0,2,2,4,-4,-2,-2,0,-2,0,2,
%U 2,-2,0,0,2,0,2,0,7,-5,-3,-5,-1,-5,-3,3,1,-5,-3,1
%N Skew-alternating sum of the reversed n-th composition in standard order.
%C 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 + ....
%C 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.
%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e The 357-th composition is (2,1,3,2,1) so a(357) = 1 - 2 - 3 + 2 + 1 = -1.
%e The 358-th composition is (2,1,3,1,2) so a(358) = 2 - 1 - 3 + 1 + 2 = 1.
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t Table[skats[Reverse[stc[n]]],{n,0,100}]
%Y See link for sequences related to standard compositions.
%Y The half-alternating form is A357622, non-reverse A357621.
%Y The reverse version is A357623.
%Y Positions of zeros are A357628, non-reverse A357627.
%Y The version for prime indices is A357630.
%Y The version for Heinz numbers of partitions is A357634.
%Y A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y Cf. A001700, A001511, A053251, A344619, A357136, A357182, A357183, A357184, A357185, A357625, A357626, A357629, A357640.
%K sign
%O 0,3
%A _Gus Wiseman_, Oct 08 2022