A233932 Irregular table read by rows: T(n,k) is the binary representation of n shifted right k times and incremented if the last bit shifted away was set.

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

Offset: 1

Author

Ralf Stephan, Dec 18 2013

Keywords

• nonn
• tabf

Comments

The last bit shifted away is the (k-1)-th bit from the right.

The length of the n-th row is A070939(n).

Terms in the n-th row add to n.

From Gary W. Adamson, Jun 07 2021: (Start)

A production matrix is the irregular triangle (n, A001511(k)) = 1 otherwise 0:

1;

0, 1;

1;

0, 0, 1;

1;

0, 1;

1;

0, 0, 0, 1;

...

Let the matrix = M, then take partial sums from the top -> down, by columns.

The result is A233932. If the 1's are replaced by any sequence, say 2n-1, the result using the same procedure is an analogous sequence with row sums equal to the partial sums of the replacement sequence. For example, with (1, 3, 5, 7, ...) the result is an array with row sums = the squares, (1, 4, 9, 16, 25, ...), as follows:

1;

1, 3;

6, 3;

6, 3, 7;

15, 3, 7;

15, 14, 7;

28, 14, 7;

28, 14, 7, 15;

45, 14, 7, 15;

45, 37, 7, 15;

... (End)

Examples

			22 in binary is 10110, so the row length is 5. T(22, 1) = 11, T(22, 2) = 5 + 1 = 6, T(22, 3) = 2 + 1 = 3, T(22, 4) = 1, T(22, 5) = 0 + 1. So the 22nd row reads 11, 6, 3, 1, 1.
Table starts:
  1;
  1, 1;
  2, 1;
  2, 1, 1;
  3, 1, 1;
  3, 2, 1;
  4, 2, 1;
  4, 2, 1, 1;
  5, 2, 1, 1;
  5, 3, 1, 1;
  ...
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A120385.

Cf. A079559.

Programs

PARI

T(n,k)=b=binary(n);n\2^k+b[#b-k+1]

PARI

row(n) = my(b=binary(n)); vector(#b, k, n\2^k+b[#b-k+1]); \\ Michel Marcus, Sep 03 2019

Formula

T(n,k) = round(n/2^k), 1 <= k <= floor(log_2(n)) + 1, where round(1/2)=1. - Ridouane Oudra, Sep 02 2019