cp's OEIS Frontend

A357622 Half-alternating sum of the reversed n-th composition in standard order.

%I A357622 #9 Oct 08 2022 09:42:10
%S A357622 0,1,2,2,3,3,3,1,4,4,4,0,4,2,2,0,5,5,5,-1,5,1,1,-1,5,3,3,-1,3,1,1,1,6,
%T A357622 6,6,-2,6,0,0,-2,6,2,2,-2,2,0,0,2,6,4,4,-2,4,0,0,0,4,2,2,0,2,2,2,2,7,
%U A357622 7,7,-3,7,-1,-1,-3,7,1,1,-3,1,-1,-1,3,7,3
%N A357622 Half-alternating sum of the reversed n-th composition in standard order.
%C A357622 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 A357622 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 A357622 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 A357622 The 357-th composition is (2,1,3,2,1) so a(357) = 1 + 2 - 3 - 1 + 2 = 1.
%t A357622 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357622 halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]),{i,Length[f]}];
%t A357622 Table[halfats[Reverse[stc[n]]],{n,0,100}]
%Y A357622 See link for sequences related to standard compositions.
%Y A357622 This is the reverse version of A357621.
%Y A357622 The skew-alternating form is A357624, non-reverse A357623.
%Y A357622 Positions of zeros are A357626, reverse A357625.
%Y A357622 The version for prime indices is A357629.
%Y A357622 The version for Heinz numbers of partitions is A357633.
%Y A357622 A124754 gives alternating sum of standard compositions, reverse A344618.
%Y A357622 A357637 counts partitions by half-alternating sum, skew A357638.
%Y A357622 A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.
%Y A357622 Cf. A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357627, A357628, A357631, A357635.
%K A357622 sign
%O A357622 0,3
%A A357622 _Gus Wiseman_, Oct 08 2022