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

Author

Franklin T. Adams-Watters, Nov 06 2006

Keywords

• nonn
• easy
• look
• tabf

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

Examples

			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
		

Links

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

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.

Programs

Mathematica

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

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).