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A357628 Numbers k such that the reversed k-th composition in standard order has skew-alternating sum 0.

%I A357628 #5 Oct 08 2022 14:16:15
%S A357628 0,3,10,14,15,36,43,44,45,52,54,58,59,61,63,136,147,149,152,153,166,
%T A357628 168,170,175,178,179,181,183,185,190,200,204,211,212,213,217,219,221,
%U A357628 228,230,234,235,237,239,242,246,247,250,254,255,528,547,549,553,560
%N A357628 Numbers k such that the reversed k-th composition in standard order has skew-alternating sum 0.
%C A357628 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 A357628 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 A357628 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 A357628 The sequence together with the corresponding compositions begins:
%e A357628     0: ()
%e A357628     3: (1,1)
%e A357628    10: (2,2)
%e A357628    14: (1,1,2)
%e A357628    15: (1,1,1,1)
%e A357628    36: (3,3)
%e A357628    43: (2,2,1,1)
%e A357628    44: (2,1,3)
%e A357628    45: (2,1,2,1)
%e A357628    52: (1,2,3)
%e A357628    54: (1,2,1,2)
%e A357628    58: (1,1,2,2)
%e A357628    59: (1,1,2,1,1)
%e A357628    61: (1,1,1,2,1)
%e A357628    63: (1,1,1,1,1,1)
%t A357628 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357628 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357628 Select[Range[0,100],skats[Reverse[stc[#]]]==0&]
%Y A357628 See link for sequences related to standard compositions.
%Y A357628 The alternating form is A344619.
%Y A357628 Positions of zeros are A357624, non-reverse A357623.
%Y A357628 The half-alternating form is A357626, non-reverse A357625.
%Y A357628 The non-reverse version is A357627.
%Y A357628 The version for prime indices is A357632.
%Y A357628 The version for Heinz numbers of partitions is A357636.
%Y A357628 A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357628 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357628 A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y A357628 Cf. A001700, A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357622, A357635, A357640.
%K A357628 nonn
%O A357628 1,2
%A A357628 _Gus Wiseman_, Oct 08 2022