A389736 Standard composition numbers of integer compositions whose 0-prepended first differences are not all equal.

3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset

1,1

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.

Links

Table of n, a(n) for n=1..66.

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !SameQ@@Differences[Prepend[stc[#], 0]]&]

Crossrefs

Subsets not of this type are counted by A054519.

For partitions instead of compositions we have complement of A325327.

For distinct non 0-prepended differences we have A389598, complement A389597.

The complement is A389732, counted by A007862.

The non 0-prepended version is A389735, counted by A389741.

For distinct instead of equal differences we have A389738, complement A389734.

These compositions are counted by A389742.

A000120 gives length of standard compositions.

A011782 counts compositions.

A066099 lists compositions in standard order.

A070939 gives sum of standard compositions.

A124767 counts maximal runs in standard compositions, anti-runs A333381.

A358133 lists first differences of standard compositions, 0-prepended A389733.

Sequence in context: A047747 A007957 A194377 * A344449 A258432 A344000

Adjacent sequences: A389733 A389734 A389735 * A389737 A389738 A389739

Keyword

nonn

Author

Gus Wiseman, Oct 15 2025

Status

approved