cp's OEIS Frontend

A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.

%I A357184 #13 Sep 29 2022 12:57:02
%S A357184 0,1,9,19,22,28,34,69,74,84,104,132,135,141,153,177,225,265,271,274,
%T A357184 283,286,292,307,310,316,328,355,358,364,376,400,451,454,460,472,496,
%U A357184 520,523,526,533,538,553,562,593,610,673,706,833,898,1041,1047,1053,1058
%N A357184 Numbers k such that the k-th composition in standard order has the same length as its alternating sum.
%C A357184 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 A357184 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%H A357184 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 A357184 The sequence together with the corresponding compositions begins:
%e A357184     0: ()
%e A357184     1: (1)
%e A357184     9: (3,1)
%e A357184    19: (3,1,1)
%e A357184    22: (2,1,2)
%e A357184    28: (1,1,3)
%e A357184    34: (4,2)
%e A357184    69: (4,2,1)
%e A357184    74: (3,2,2)
%e A357184    84: (2,2,3)
%e A357184   104: (1,2,4)
%e A357184   132: (5,3)
%e A357184   135: (5,1,1,1)
%e A357184   141: (4,1,2,1)
%e A357184   153: (3,1,3,1)
%e A357184   177: (2,1,4,1)
%e A357184   225: (1,1,5,1)
%t A357184 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357184 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A357184 Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]
%Y A357184 See link for sequences related to standard compositions.
%Y A357184 For product equal to sum we have A335404, counted by A335405.
%Y A357184 For sum equal to twice alternating sum we have A348614, counted by A262977.
%Y A357184 These compositions are counted by A357182.
%Y A357184 For absolute value we have A357184, counted by A357183.
%Y A357184 The case of partitions is counted by A357189.
%Y A357184 A003242 counts anti-run compositions, ranked by A333489.
%Y A357184 A011782 counts compositions.
%Y A357184 A025047 counts alternating compositions, ranked by A345167.
%Y A357184 A032020 counts strict compositions, ranked by A233564.
%Y A357184 A124754 gives alternating sums of standard compositions.
%Y A357184 A238279 counts compositions by sum and number of maximal runs.
%Y A357184 A357136 counts compositions by alternating sum.
%Y A357184 Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.
%K A357184 nonn
%O A357184 1,3
%A A357184 _Gus Wiseman_, Sep 28 2022