A390747 Numbers k such that there is no nonnegative integer sequence whose first sums are the k-th composition in standard order.

25, 49, 51, 57, 81, 89, 97, 98, 99, 102, 103, 109, 113, 115, 121, 153, 161, 163, 177, 179, 185, 193, 194, 195, 197, 198, 199, 204, 205, 206, 207, 209, 217, 219, 225, 226, 227, 230, 231, 237, 241, 243, 249, 281, 289, 305, 307, 313, 321, 322, 323, 326, 327, 333

Offset

1,1

Comments

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

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.

Also integers k such that for no b are the b-based partial alternating sums of the k-th composition in standard order all nonnegative.

Links

Table of n, a(n) for n=1..54.

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

Example

For n = 24 there are two nonnegative sequences whose first sums are (1,4), namely (0,1,3) and (1,0,4), so 24 is not in the sequence.
For n = 25 there are no nonnegative sequences whose first sums are (1,3,1), so 25 is in the sequence.
For n = 52 there are two nonnegative sequences whose first sums are (1,2,3), namely (0,1,1,2) and (1,0,2,1), so 52 is not in the sequence.
For n = 150, all sequences with first sums (3,2,1,2) are of the form (b,3-b,-1+b,2-b,b) for some b. This is nonnegative for b = 1 or b = 2, so 150 is not in the sequence.
The terms together with the corresponding standard compositions begin:
   25: (1,3,1)
   49: (1,4,1)
   51: (1,3,1,1)
   57: (1,1,3,1)
   81: (2,4,1)
   89: (2,1,3,1)
   97: (1,5,1)
   98: (1,4,2)
   99: (1,4,1,1)
  102: (1,3,1,2)
  103: (1,3,1,1,1)
  109: (1,2,1,2,1)
  113: (1,1,4,1)
  115: (1,1,3,1,1)
  121: (1,1,1,3,1)
  153: (3,1,3,1)
  161: (2,5,1)
  163: (2,4,1,1)
  177: (2,1,4,1)
  179: (2,1,3,1,1)
  185: (2,1,1,3,1)

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
pas[y_, k_]:=Table[(-1)^j*k+Sum[(-1)^(i-j)*y[[i]], {i, j}], {j, 0, Length[y]}];
Select[Range[100], Select[Table[pas[stc[#], b], {b, 0, Max[stc[#]]}], Min@@#>=0&]=={}&]

Crossrefs

The complement for compositions is A390676 = union of A390568.

For positive instead of nonnegative sequences we have A390677.

For a unique choice of compositions we have A390745, counted by A391644.

These are positions of 0 in A391621.

For a unique choice we have A391622, counted by A391643.

For more than one choice we have A391623.

These compositions are counted by A391645.

A011782 counts compositions.

A066099 lists all compositions in standard order.

A357213 counts compositions by sum of first sums.

A390432 lists first sums of standard compositions.

A390449 ranks first sums of prime indices, listed by A390307 or A390362.

A390673 ranks compositions with all distinct first sums, counted by A390567.

A390674 ranks compositions with all equal first sums, counted by A342527.

Cf. A000120, A001511, A029837, A029931, A070939, A389731, A390448, A390675, A390678.

Sequence in context: A004936 A062058 A273510 * A198591 A069063 A064937

Adjacent sequences: A390744 A390745 A390746 * A390748 A390749 A390750

Keyword

nonn

Author

Gus Wiseman, Dec 19 2025

Status

approved