A055941 a(n) = Sum_{j=0..k-1} (i(j) - j) where n = Sum_{j=0..k-1} 2^i(j).

0, 0, 1, 0, 2, 1, 2, 0, 3, 2, 3, 1, 4, 2, 3, 0, 4, 3, 4, 2, 5, 3, 4, 1, 6, 4, 5, 2, 6, 3, 4, 0, 5, 4, 5, 3, 6, 4, 5, 2, 7, 5, 6, 3, 7, 4, 5, 1, 8, 6, 7, 4, 8, 5, 6, 2, 9, 6, 7, 3, 8, 4, 5, 0, 6, 5, 6, 4, 7, 5, 6, 3, 8, 6, 7, 4, 8, 5, 6, 2, 9, 7, 8, 5, 9, 6, 7, 3, 10, 7, 8, 4, 9, 5, 6, 1, 10, 8, 9, 6, 10, 7, 8, 4

Offset: 0

Author

Anno Siegel (siegel(AT)zrz.tu-berlin.de), Jul 18 2000

Keywords

• nonn
• base

Comments

Used to calculate number of subspaces of Zp^n where Zp is field of integers mod p.

Consider a square matrix A and call it special if (0) A is an upper triangular matrix, (1) a nonzero column of A has a 1 on the main diagonal and (2) if a row has a 1 on the main diagonal then this is the only nonzero element in that row.

If the diagonal of a special matrix is given (it can only contain 0's and 1's), many of the fields of A are determined by (0), (1) and (2). The number of fields that can be freely chosen while still satisfying (0), (1) and (2) is a(n), where n is the diagonal, read as a binary number with least significant bit at upper left.

a(n) is also the minimum number of adjacent bit swap operations required to pack all the ones of n to the right. - Philippe Beaudoin, Aug 19 2014

From Rakesh Khanna A, Aug 06 2021: (Start)

a(n) is also the area under the curve formed from the binary representation of n where each 0-bit corresponds to an increase of one unit along the x-axis and each 1-bit corresponds to an increase of one unit along the y-axis.

E.g., n = 20 = 10100_2 and the area under the curve shown below is a(n) = 5.

1 0 1 0 0

\ \ \ \ \ |

\ \ \+----+----+

\ \ | |

\+----+ +

| |

----+----+----+----+

(End)

Examples

			20 = 2^4 + 2^2, thus a(20) = (2-0) + (4-1) = 5.
		

References

• A. Siegel, Linear Aspects of Boolean Functions, 1999 (unpublished).

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Philip Lafrance, Narad Rampersad and Randy Yee, Some properties of a Rudin-Shapiro-like sequence, arXiv:1408.2277 [math.CO], 2014 (see page 2).

Crossrefs

Cf. A000120, A029837, A161511, A161920, A126441.

Programs

Mathematica

b[n_] := b[n] = If[n == 0, 0, If[EvenQ[n], b[n/2] + DigitCount[n/2, 2, 1], b[(n - 1)/2] + 1]];
a[n_] := b[n] - DigitCount[n, 2, 1];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 23 2018 *)

PARI

a(n) = {my(b=binary(n)); nb = 0; for (i=1, #b-1, if (b[i], nb += sum(j=i+1, #b, !b[j]));); nb;} \\ Michel Marcus, Aug 12 2014

Python

def A055941(n):
    s = bin(n)[2:]
    return sum(s[i:].count('0') for i,d in enumerate(s,start=1) if d == '1')
# Chai Wah Wu, Sep 07 2014

Formula

a(n) = Sum (total number of 0-bits to the right of 1-bit) over all 1-bits of n.

a(n) = A161511(n) - A000120(n) = A161920(n+1) - 1 - A029837(n+1).

a(n) = 0 if A241816(n) = n; 1 + a(A241816(n)) otherwise. - Philippe Beaudoin, Aug 19 2014

Extensions

Edited and extended by Antti Karttunen, Oct 12 2009