A357628 Numbers k such that the reversed k-th composition in standard order has skew-alternating sum 0.

0, 3, 10, 14, 15, 36, 43, 44, 45, 52, 54, 58, 59, 61, 63, 136, 147, 149, 152, 153, 166, 168, 170, 175, 178, 179, 181, 183, 185, 190, 200, 204, 211, 212, 213, 217, 219, 221, 228, 230, 234, 235, 237, 239, 242, 246, 247, 250, 254, 255, 528, 547, 549, 553, 560

Offset

1,2

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

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.

Links

Table of n, a(n) for n=1..55.

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

Example

The sequence together with the corresponding compositions begins:
    0: ()
    3: (1,1)
   10: (2,2)
   14: (1,1,2)
   15: (1,1,1,1)
   36: (3,3)
   43: (2,2,1,1)
   44: (2,1,3)
   45: (2,1,2,1)
   52: (1,2,3)
   54: (1,2,1,2)
   58: (1,1,2,2)
   59: (1,1,2,1,1)
   61: (1,1,1,2,1)
   63: (1,1,1,1,1,1)

Mathematica

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

Crossrefs

See link for sequences related to standard compositions.

The alternating form is A344619.

Positions of zeros are A357624, non-reverse A357623.

The half-alternating form is A357626, non-reverse A357625.

The non-reverse version is A357627.

The version for prime indices is A357632.

The version for Heinz numbers of partitions is A357636.

A124754 gives alternating sum of standard compositions, reverse A344618.

A357637 counts partitions by half-alternating sum, skew A357638.

A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.

Cf. A001700, A001511, A053251, A357136, A357182, A357183, A357184, A357185, A357622, A357635, A357640.

Sequence in context: A002354 A079943 A041865 * A289151 A085776 A289106

Adjacent sequences: A357625 A357626 A357627 * A357629 A357630 A357631

Keyword

nonn

Author

Gus Wiseman, Oct 08 2022

Status

approved