1,1

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

Intersection of A022340 (= 2*A003714) and A390677.

For 72 we have the standard composition (3,4), which is the first sums of two compositions: (1,2,2) and (2,1,3), so 72 is not in the sequence.

For 82 we have the standard composition (2,3,2), which is not the first sums of any composition, so 82 is in the sequence.

For 658 we have the standard composition (2,3,3,2), which is the first sums of (1,1,2,1,1), so 658 is not in the sequence.

The terms together with standard compositions begin:

82: (2,3,2)

162: (2,4,2)

290: (3,4,2)

322: (2,5,2)

324: (2,4,3)

330: (2,3,2,2)

338: (2,2,3,2)

578: (3,5,2)

594: (3,2,3,2)

642: (2,6,2)

644: (2,5,3)

650: (2,4,2,2)

660: (2,3,2,3)

674: (2,2,4,2)

1090: (4,5,2)

1106: (4,2,3,2)

1154: (3,6,2)

1156: (3,5,3)

1162: (3,4,2,2)

1186: (3,2,4,2)

1282: (2,7,2)

1284: (2,6,3)

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

uncha[tar_, ini_]:=(cur={ini}; Do[AppendTo[cur, tar[[k]]-If[k==1, ini, cur[[k]]]], {k, Length[tar]}]; cur);

Select[Range[1000], Length[Select[Table[uncha[stc[#], i], {i, Max[stc[#]]}], Min@@#>=1&]]==0&&!MemberQ[stc[#], 1]&]

These indicate certain zeros in the pre-bisected A390675.

Allowing 1's gives A390677, complement of A390676 = union of A390568.

The unique version is A390745.

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 distinct first sums, counted by A390567.

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

Cf. A000079, A000120, A001511, A003714, A008965, A022340, A029837, A029931, A070211, A070939, A389731, A390448.

Sequence in context: A260835 A173087 A264352 * A044252 A044633 A158123

Adjacent sequences: A390675 A390676 A390677 * A390679 A390680 A390681

Gus Wiseman, Nov 23 2025

approved