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A357627 Numbers k such that the k-th composition in standard order has skew-alternating sum 0.

%I A357627 #5 Oct 08 2022 14:16:11
%S A357627 0,3,10,11,15,36,37,38,43,45,54,55,58,59,63,136,137,138,140,147,149,
%T A357627 153,166,167,170,171,175,178,179,183,190,191,204,205,206,212,213,214,
%U A357627 219,221,228,229,230,235,237,246,247,250,251,255,528,529,530,532,536
%N A357627 Numbers k such that the k-th composition in standard order has skew-alternating sum 0.
%C A357627 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 A357627 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 A357627 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 A357627 The sequence together with the corresponding compositions begins:
%e A357627     0: ()
%e A357627     3: (1,1)
%e A357627    10: (2,2)
%e A357627    11: (2,1,1)
%e A357627    15: (1,1,1,1)
%e A357627    36: (3,3)
%e A357627    37: (3,2,1)
%e A357627    38: (3,1,2)
%e A357627    43: (2,2,1,1)
%e A357627    45: (2,1,2,1)
%e A357627    54: (1,2,1,2)
%e A357627    55: (1,2,1,1,1)
%e A357627    58: (1,1,2,2)
%e A357627    59: (1,1,2,1,1)
%e A357627    63: (1,1,1,1,1,1)
%t A357627 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357627 skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
%t A357627 Select[Range[0,100],skats[stc[#]]==0&]
%Y A357627 See link for sequences related to standard compositions.
%Y A357627 The alternating form is A344619.
%Y A357627 Positions of zeros in A357623.
%Y A357627 The half-alternating form is A357625, reverse A357626.
%Y A357627 The reverse version is A357628.
%Y A357627 The version for prime indices is A357632.
%Y A357627 The version for Heinz numbers of partitions is A357636.
%Y A357627 A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357627 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357627 A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y A357627 Cf. A001700, A001511, A053251, A357136, A357182-A357185, A357621, A357624, A357630, A357634, A357640.
%K A357627 nonn
%O A357627 1,2
%A A357627 _Gus Wiseman_, Oct 08 2022