cp's OEIS Frontend

A357626 Numbers k such that the reversed k-th composition in standard order has half-alternating sum 0.

%I A357626 #6 Oct 08 2022 14:16:01
%S A357626 0,11,15,37,38,45,46,53,54,55,59,137,138,140,153,154,156,167,169,170,
%T A357626 171,172,179,191,201,202,204,205,206,213,214,229,230,231,235,243,247,
%U A357626 251,255,529,530,532,536,561,562,564,568,583,587,593,594,595,596,600
%N A357626 Numbers k such that the reversed k-th composition in standard order has half-alternating sum 0.
%C A357626 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 A357626 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 A357626 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 A357626 The sequence together with the corresponding compositions begins:
%e A357626     0: ()
%e A357626    11: (2,1,1)
%e A357626    15: (1,1,1,1)
%e A357626    37: (3,2,1)
%e A357626    38: (3,1,2)
%e A357626    45: (2,1,2,1)
%e A357626    46: (2,1,1,2)
%e A357626    53: (1,2,2,1)
%e A357626    54: (1,2,1,2)
%e A357626    55: (1,2,1,1,1)
%e A357626    59: (1,1,2,1,1)
%t A357626 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357626 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357626 Select[Range[0,100],halfats[Reverse[stc[#]]]==0&]
%Y A357626 See link for sequences related to standard compositions.
%Y A357626 The alternating form is A344619.
%Y A357626 Positions of zeros in A357622.
%Y A357626 The non-reverse version is A357625.
%Y A357626 The skew-alternating form is A357628, reverse A357627.
%Y A357626 The version for prime indices is A357631.
%Y A357626 The version for Heinz numbers of partitions is A357635.
%Y A357626 A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357626 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357626 A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y A357626 Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357621, A357623, A357629, A357633.
%K A357626 nonn
%O A357626 1,2
%A A357626 _Gus Wiseman_, Oct 08 2022