cp's OEIS Frontend

A357185 Numbers k such that the k-th composition in standard order has the same length as the absolute value of its alternating sum.

%I A357185 #10 Sep 29 2022 12:57:51
%S A357185 0,1,9,12,19,22,28,34,40,69,74,84,97,104,132,135,141,144,153,177,195,
%T A357185 198,204,216,225,240,265,271,274,283,286,292,307,310,316,321,328,355,
%U A357185 358,364,376,386,400,451,454,460,472,496,520,523,526,533,538,544,553
%N A357185 Numbers k such that the k-th composition in standard order has the same length as the absolute value of its alternating sum.
%C A357185 A composition of n is a finite sequence of positive integers summing to n. 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.
%C A357185 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%H A357185 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 A357185 The sequence together with the corresponding compositions begins:
%e A357185     0: ()
%e A357185     1: (1)
%e A357185     9: (3,1)
%e A357185    12: (1,3)
%e A357185    19: (3,1,1)
%e A357185    22: (2,1,2)
%e A357185    28: (1,1,3)
%e A357185    34: (4,2)
%e A357185    40: (2,4)
%e A357185    69: (4,2,1)
%e A357185    74: (3,2,2)
%e A357185    84: (2,2,3)
%e A357185    97: (1,5,1)
%e A357185   104: (1,2,4)
%e A357185   132: (5,3)
%e A357185   135: (5,1,1,1)
%e A357185   141: (4,1,2,1)
%t A357185 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357185 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357185 Select[Range[0,100],Length[stc[#]]==Abs[ats[stc[#]]]&]
%Y A357185 See link for sequences related to standard compositions.
%Y A357185 For sum equal to twice alternating sum we have A348614, counted by A262977.
%Y A357185 For product equal to sum we have A335404, counted by A335405.
%Y A357185 These compositions are counted by A357183.
%Y A357185 This is the absolute value version of A357184, counted by A357183.
%Y A357185 A003242 counts anti-run compositions, ranked by A333489.
%Y A357185 A011782 counts compositions.
%Y A357185 A025047 counts alternating compositions, ranked by A345167.
%Y A357185 A032020 counts strict compositions, ranked by A233564.
%Y A357185 A124754 gives alternating sums of standard compositions.
%Y A357185 A238279 counts compositions by sum and number of maximal runs.
%Y A357185 A357136 counts compositions by alternating sum.
%Y A357185 Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.
%K A357185 nonn
%O A357185 1,3
%A A357185 _Gus Wiseman_, Sep 28 2022