A390090 Irregular triangle read by rows where row n lists the first differences of the binary indices of n.

1, 2, 1, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 1, 3, 2, 2, 2, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 4, 3, 2, 3, 1, 3, 1, 1, 3, 2, 3, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 3, 1

Offset

3,2

Comments

Rows 0, 1, 2, are empty, so offset is 3.

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 27 are {1,2,4,5}, so row n = 27 is (1,2,1).
Triangle begins:
   0:         16:           32:
   1:         17: 4         33: 5
   2:         18: 3         34: 4
   3: 1       19: 1 3       35: 1 4
   4:         20: 2         36: 3
   5: 2       21: 2 2       37: 2 3
   6: 1       22: 1 2       38: 1 3
   7: 1 1     23: 1 1 1     39: 1 1 3
   8:         24: 1         40: 2
   9: 3       25: 3 1       41: 3 2
  10: 2       26: 2 1       42: 2 2
  11: 1 2     27: 1 2 1     43: 1 2 2
  12: 1       28: 1 1       44: 1 2
  13: 2 1     29: 2 1 1     45: 2 1 2
  14: 1 1     30: 1 1 1     46: 1 1 2
  15: 1 1 1   31: 1 1 1 1   47: 1 1 1 2

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Differences[bpe[n]], {n, 100}]

Crossrefs

Positions of empty rows are A000079.

Rows of length 1 are indicated by A018900.

Rows of length 0 or 1 are indicated by A048645.

Row lengths are A048881.

Row sums are A119387.

The 0-prepended version is A228351.

Constant rows are indicated by A295235.

Maxima of odd-indexed rows are A333766.

Minima of odd-indexed rows are A333768.

For prime indices we have A355536, ranks A253566.

For standard compositions we have A358133, row sums A358135.

Strict rows are indicated by A390091.

A011782 counts compositions.

A066099 lists all compositions in standard order, reverse A228351.

A325545 counts compositions with distinct first differences.

Cf. A000120, A001511, A029837, A029931, A070211, A070939, A175332, A325352, A325546, A390432.

Sequence in context: A344756 A101221 A086291 * A262766 A262747 A016442

Adjacent sequences: A390087 A390088 A390089 * A390091 A390092 A390093

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 19 2025

Status

approved