1,1

A finite arithmetic progression is a sequence with all equal first differences.

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 terms together with corresponding standard compositions begin:

11: (2,1,1)

13: (1,2,1)

14: (1,1,2)

19: (3,1,1)

21: (2,2,1)

22: (2,1,2)

23: (2,1,1,1)

25: (1,3,1)

26: (1,2,2)

27: (1,2,1,1)

28: (1,1,3)

29: (1,1,2,1)

30: (1,1,1,2)

35: (4,1,1)

38: (3,1,2)

39: (3,1,1,1)

41: (2,3,1)

43: (2,2,1,1)

44: (2,1,3)

45: (2,1,2,1)

46: (2,1,1,2)

47: (2,1,1,1,1)

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

Select[Range[0, 100], !SameQ@@Differences[stc[#]]&]

For partitions instead of compositions we have complement of A325328.

For distinct instead of equal differences we have A389598, complement A389597.

The complement is A389731, counted by A175342.

The 0-prepended version is A389736, counted by A389742.

These compositions are counted by A389741.

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.

Cf. A332834, A375408.

Sequence in context: A111347 A026818 A385411 * A163140 A048596 A228709

Adjacent sequences: A389732 A389733 A389734 * A389736 A389737 A389738

Gus Wiseman, Oct 14 2025

approved