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

0, 1, 9, 19, 22, 28, 34, 69, 74, 84, 104, 132, 135, 141, 153, 177, 225, 265, 271, 274, 283, 286, 292, 307, 310, 316, 328, 355, 358, 364, 376, 400, 451, 454, 460, 472, 496, 520, 523, 526, 533, 538, 553, 562, 593, 610, 673, 706, 833, 898, 1041, 1047, 1053, 1058

Offset: 1

Author

Gus Wiseman, Sep 28 2022

Keywords

• nonn

Comments

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.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The sequence together with the corresponding compositions begins:
    0: ()
    1: (1)
    9: (3,1)
   19: (3,1,1)
   22: (2,1,2)
   28: (1,1,3)
   34: (4,2)
   69: (4,2,1)
   74: (3,2,2)
   84: (2,2,3)
  104: (1,2,4)
  132: (5,3)
  135: (5,1,1,1)
  141: (4,1,2,1)
  153: (3,1,3,1)
  177: (2,1,4,1)
  225: (1,1,5,1)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

For product equal to sum we have A335404, counted by A335405.

For sum equal to twice alternating sum we have A348614, counted by A262977.

These compositions are counted by A357182.

For absolute value we have A357184, counted by A357183.

The case of partitions is counted by A357189.

A003242 counts anti-run compositions, ranked by A333489.

A011782 counts compositions.

A025047 counts alternating compositions, ranked by A345167.

A032020 counts strict compositions, ranked by A233564.

A124754 gives alternating sums of standard compositions.

A238279 counts compositions by sum and number of maximal runs.

A357136 counts compositions by alternating sum.

Cf. A000120, A070939, A114220, A114901, A178470, A242882, A262046, A301987.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],Length[stc[#]]==ats[stc[#]]&]