A348614 Numbers k such that the k-th composition in standard order has sum equal to twice its alternating sum.

0, 9, 11, 14, 130, 133, 135, 138, 141, 143, 148, 153, 155, 158, 168, 177, 179, 182, 188, 208, 225, 227, 230, 236, 248, 2052, 2057, 2059, 2062, 2066, 2069, 2071, 2074, 2077, 2079, 2084, 2089, 2091, 2094, 2098, 2101, 2103, 2106, 2109, 2111, 2120, 2129, 2131

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..48.

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

Example

The terms together with their binary indices begin:
    0: ()
    9: (3,1)
   11: (2,1,1)
   14: (1,1,2)
  130: (6,2)
  133: (5,2,1)
  135: (5,1,1,1)
  138: (4,2,2)
  141: (4,1,2,1)
  143: (4,1,1,1,1)
  148: (3,2,3)
  153: (3,1,3,1)
  155: (3,1,2,1,1)
  158: (3,1,1,1,2)

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 1000], Total[stc[#]]==2*ats[stc[#]]&]

Crossrefs

The unordered case (partitions) is counted by A000712, reverse A006330.

These compositions are counted by A262977.

Except for 0, a subset of A345917 (which is itself a subset of A345913).

A000346 = even-length compositions with alt sum != 0, complement A001700.

A011782 counts compositions.

A025047 counts wiggly compositions, ranked by A345167.

A034871 counts compositions of 2n with alternating sum 2k.

A097805 counts compositions by alternating (or reverse-alternating) sum.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A116406 counts compositions with alternating sum >=0, ranked by A345913.

A138364 counts compositions with alternating sum 0, ranked by A344619.

A345197 counts compositions by length and alternating sum.

Cf. A000984, A002458, A004331, A005810, A224274.

Cf. A008549, A013777, A027306, A058622, A088218, A114121, A120452, A126869, A163493, A294175, A344604.

Sequence in context: A212816 A183980 A101754 * A113339 A270190 A320701

Adjacent sequences: A348611 A348612 A348613 * A348615 A348616 A348617

Keyword

nonn

Author

Gus Wiseman, Oct 29 2021

Status

approved