A228368 Difference between the n-th element of the ruler function and the highest power of 2 dividing n.

0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -11, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -26, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -11, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -57, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -11, 0, 0, 0, -1, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, -26

Offset: 1

Author

Omar E. Pol, Aug 22 2013

Keywords

• sign
• tabf

Comments

Also rank of the n-th region of the diagram of compositions of j, if 1 <= n <= 2^(j-1), see example.

Here the rank of a region is defined as the largest part minus the number of parts (similar to the Dyson's rank of a partition).

The equivalent sequence for integer partitions is A194447.

Also triangle read by rows in which T(j,k) is the rank of the k-th region of the j-th section of the set of compositions in colexicographic order of any integer >= j. See A228366.

Examples

			Illustration of initial terms (n = 1..16):
-----------------------------------------------
.                  Largest     Number of
.    Diagram of    part of     parts of
.   compositions   region n    region n
-----------------------------------------------
n                 A001511(n)  A006519(n)  a(n)
-----------------------------------------------
.
1     _| | | | |      1           1         0
2     _ _| | | |      2           2         0
3     _|   | | |      1           1         0
4     _ _ _| | |      3           4        -1
5     _| |   | |      1           1         0
6     _ _|   | |      2           2         0
7     _|     | |      1           1         0
8     _ _ _ _| |      4           8        -4
9     _| | |   |      1           1         0
10    _ _| |   |      2           2         0
11    _|   |   |      1           1         0
12    _ _ _|   |      3           4        -1
13    _| |     |      1           1         0
14    _ _|     |      2           2         0
15    _|       |      1           1         0
16    _ _ _ _ _|      5          16       -11
.
Written as an array read by rows with four columns the first three columns contain only zeros.
  0,   0,   0,  -1;
  0,   0,   0,  -4;
  0,   0,   0,  -1;
  0,   0,   0, -11;
  0,   0,   0,  -1;
  0,   0,   0,  -4;
  0,   0,   0,  -1;
  0,   0,   0, -26;
  ...
Written as a triangle T(j,k) the sequence begins:
  0;
  0;
  0,-1;
  0,0,0,-4;
  0,0,0,-1,0,0,0,-11;
  0,0,0,-1,0,0,0,-4,0,0,0,-1,0,0,0,-26;
  0,0,0,-1,0,0,0,-4,0,0,0,-1,0,0,0,-11,0,0,0,-1,0,0,0,-4,0, 0,0,-1,0,0,0,-57;
  ...
Row lengths give A011782.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A001511, A006519, A011782, A141285, A194446, A194447, A228366, A228367, A228525.

Programs

Python

def A228368(n): return (m:=n&-n).bit_length()-m # Chai Wah Wu, Jul 14 2022

Formula

a(n) = A001511(n) - A006519(n).

a(4n-3) = a(4n-2) = a(4n-1) = 0. a(4n) = A001511(4n) - A006519(4n).