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%I #6 Oct 08 2022 14:16:01
%S 0,11,15,37,38,45,46,53,54,55,59,137,138,140,153,154,156,167,169,170,
%T 171,172,179,191,201,202,204,205,206,213,214,229,230,231,235,243,247,
%U 251,255,529,530,532,536,561,562,564,568,583,587,593,594,595,596,600
%N Numbers k such that the reversed k-th composition in standard order has half-alternating sum 0.
%C 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 - ...
%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 sequence together with the corresponding compositions begins:
%e 0: ()
%e 11: (2,1,1)
%e 15: (1,1,1,1)
%e 37: (3,2,1)
%e 38: (3,1,2)
%e 45: (2,1,2,1)
%e 46: (2,1,1,2)
%e 53: (1,2,2,1)
%e 54: (1,2,1,2)
%e 55: (1,2,1,1,1)
%e 59: (1,1,2,1,1)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t Select[Range[0,100],halfats[Reverse[stc[#]]]==0&]
%Y See link for sequences related to standard compositions.
%Y The alternating form is A344619.
%Y Positions of zeros in A357622.
%Y The non-reverse version is A357625.
%Y The skew-alternating form is A357628, reverse A357627.
%Y The version for prime indices is A357631.
%Y The version for Heinz numbers of partitions is A357635.
%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. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357621, A357623, A357629, A357633.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 08 2022
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