A124757 Zero-based weighted sum of compositions in standard order.

0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 3, 4, 5, 6, 0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 0, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 7, 8, 9, 10, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14

Offset

0,7

Comments

The standard order of compositions is given by A066099.

Sum of all positions of 1's except the last in the reversed binary expansion of n. For example, the reversed binary expansion of 14 is (0,1,1,1), so a(14) = 2 + 3 = 5. Keeping the last position gives A029931. - Gus Wiseman, Jan 17 2023

Links

Alois P. Heinz, Rows n = 0..14, flattened

Formula

For a composition b(1),...,b(k), a(n) = Sum_{i=1..k} (i-1)*b(i).

For n>0, a(n) = A029931(n) - A070939(n).

Example

Composition number 11 is 2,1,1; 0*2+1*1+2*1 = 3, so a(11) = 3.
The table starts:
  0
  0
  0 1
  0 1 2 3

Mathematica

Table[Total[Most[Join@@Position[Reverse[IntegerDigits[n, 2]], 1]]], {n, 30}]

Crossrefs

Cf. A066099, A070939, A029931, A011782 (row lengths), A001788 (row sums).

Row sums of A048793 if we delete the last part of every row.

For prime indices instead of standard comps we have A359674, rev A359677.

Positions of first appearances are A359756.

A003714 lists numbers with no successive binary indices.

A030190 gives binary expansion, reverse A030308.

A230877 adds up positions of 1's in binary expansion, length A000120.

A359359 adds up positions of 0's in binary expansion, length A023416.

Cf. A059015, A065359, A069010, A073642, A083652, A359400, A359402, A359678.

Sequence in context: A277904 A132387 A309954 * A242305 A049263 A014588

Adjacent sequences: A124754 A124755 A124756 * A124758 A124759 A124760

Keyword

nonn,easy,look,tabf

Author

Franklin T. Adams-Watters, Nov 06 2006

Status

approved