A390987 Irregular triangle read by rows where row k lists the first sums of binary indices of n.

3, 4, 5, 3, 5, 5, 6, 3, 6, 7, 4, 7, 5, 7, 3, 5, 7, 6, 7, 3, 7, 8, 4, 8, 5, 8, 3, 5, 8, 9, 5, 9, 6, 9, 3, 6, 9, 7, 9, 4, 7, 9, 5, 7, 9, 3, 5, 7, 9, 7, 8, 3, 8, 9, 4, 9, 5, 9, 3, 5, 9, 10, 5, 10, 6, 10, 3, 6, 10, 7, 10, 4, 7, 10, 5, 7, 10, 3, 5, 7, 10, 11, 6, 11, 7, 11

Offset

3,1

Comments

The first 3 rows are empty, so offset is 3.

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

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Links

Table of n, a(n) for n=3..88.

Example

The binary indices of 300 are (3,4,6,9), with first sums (7,10,15), which is row 300.
Triangle begins:
   0:
   1:
   2:
   3: 3
   4:
   5: 4
   6: 5
   7: 3 5
   8:
   9: 5
  10: 6
  11: 3 6
  12: 7
  13: 4 7
  14: 5 7
  15: 3 5 7

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];
Table[firsums[bpe[n]], {n, 0, 100}]

Crossrefs

First sums of A048793 (binary indices).

Row lengths are A048881.

Empty rows are indicated by A094373.

For first differences instead of first sums we have A390090.

For prime indices instead of binary indices we have A390307, ranks A390449.

For standard compositions instead of binary indices we have A390432.

Row sums are A390992.

A011782 counts compositions.

A342527 counts compositions with all equal first sums.

A390567 counts compositions with distinct first sums.

Cf. A000120, A001511, A019565, A029837, A029931, A066099, A070939, A390568.

Sequence in context: A354597 A121890 A330740 * A178231 A381191 A343262

Adjacent sequences: A390984 A390985 A390986 * A390988 A390989 A390990

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 25 2025

Status

approved