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A059015 Total number of 0's in binary expansions of 0, ..., n. 32
1, 1, 2, 2, 4, 5, 6, 6, 9, 11, 13, 14, 16, 17, 18, 18, 22, 25, 28, 30, 33, 35, 37, 38, 41, 43, 45, 46, 48, 49, 50, 50, 55, 59, 63, 66, 70, 73, 76, 78, 82, 85, 88, 90, 93, 95, 97, 98, 102, 105, 108, 110, 113, 115, 117, 118, 121, 123, 125, 126, 128, 129, 130, 130, 136, 141 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partial sums of A023416. - Reinhard Zumkeller, Jul 15 2011

The graph of this sequence is a version of the Takagi curve: see Lagarias (2012), Section 9, especially Theorem 9.1. - N. J. A. Sloane, Mar 12 2016

LINKS

T. D. Noe and Hieronymus Fischer, Table of n, a(n) for n = 0..10000 (terms up to n=1023 by T. D. Noe)

Lagarias, Jeffrey C. The Takagi function and its properties. In Functions in number theory and their probabilistic aspects, 153--189, RIMS Kôkyûroku Bessatsu, B34, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. MR3014845. Also arXiv:1112.4502.

R. Stephan, Some divide-and-conquer sequences ...

R. Stephan, Table of generating functions

Index entries for sequences related to binary expansion of n

FORMULA

a(n) = b(n)+1, with b(2n) = b(n)+b(n-1)+n, b(2n+1) = 2b(n)+n. - Ralf Stephan, Sep 13 2003

Contribution from Hieronymus Fischer, Jun 10 2012: (Start)

With m = floor(log_2(n)):

a(n) = 2 + (m+1)*(n+1) - 2^(m+1) + (1/2)*sum_{j=1..m+1} (floor(n/2^j)*(2n + 2 - (1 + floor(n/2^j))*2^j) - floor(n/2^j + 1/2)*(2n + 2 - floor(n/2^j + 1/2)*2^j.

a(n) = A083652(n) - (n+1)*A000120(n) + 2^(m-1) - (1/4) + (1/2)*sum_{j=1..m+1} (floor(n/2^j + 1/2)^2 - (floor(n/2^j) + 1/2)^2)*2^j.

a(2^m-1) = 2 + (m-2)*2^(m-1)

(this is the total number of zero digits occurring in all the numbers with <= m places).

G.f.: g(x) = 1 + (1/(1-x)^2)*sum_{j>=0} x^(2*2^j)/(1+x^2^j).

General formulas for the number of digits <= d in the base p representations of all integers from 0 to n, where 0 <= d < p.

With m = floor(log_p(n)):

a(n) = 1 + (m+1)*(n+1) - (p^(m+1)-1)/(p-1) + (1/2)*sum_{j=1..m+1} (floor(n/p^j)*(2n + 2 - (1 + floor(n/p^j))*p^j) - floor(n/p^j + (p-d-1)/p)*(2n + 2 + ((p-2*d-2)/p - floor(n/p^j + (p-d-1)/p))*p^j)).

a(n) = H(n,p) - (n+1)*F(n,p,d+1) + (1/2)*sum_{j=1..m+1} ((floor(n/p^j + (p-d-1)/p)^2 - floor(n/p^j)^2)*p^j - (((p - 2*d-2)/p)*floor(n/p^j + (p-d-1)/p) + floor(n/p^j))*p^j), where H(n,p) = sum of number of digits in the base p representations of 0 to n and F(n,p,d) = number of digits >=d in the base p representation of n.

a(p^m-1) = 1 + (d+1)*m*p^(m-1) - (p^m-1)/(p-1).

(this is the total number of digits <= d occurring in all the numbers with <= m places in base p representation).

G.f.: g(x) = 1 + (1/(1-x)^2)*sum_{j>=0} (1-x^(d*p^j))*x^p^j) + (1-x^p^j)*x^p^(j+1)/(1-x^p^(j+1)). (End)

MATHEMATICA

Accumulate[ Table[ Count[ IntegerDigits[n, 2], 0], {n, 0, 65}]] (* Jean-François Alcover, Oct 03 2012 *)

Accumulate[DigitCount[Range[0, 70], 2, 0]] (* Harvey P. Dale, Jun 24 2017 *)

PROG

(Haskell)

a059015 n = a059015_list !! n

a059015_list = scanl1 (+) $ map a023416 [0..]

-- Reinhard Zumkeller, Jul 15 2011

(PARI) v=vector(100, i, 1); for(i=1, #v-1, v[i+1] = v[i] + #binary(i) - hammingweight(i)); v \\ Charles R Greathouse IV, Nov 20 2012

(PARI) a(n)=if(n, my(m=logint(n, 2)); 2 + (m+1)*(n+1) - 2^(m+1) + sum(j=1, m+1, my(t=floor(n/2^j + 1/2)); (n>>j)*(2*n + 2 - (1 + (n>>j))<<j) - (2*n + 2 - t<<j)*t)/2, 1) \\ Charles R Greathouse IV, Dec 14 2015

CROSSREFS

The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015, A070939, A083652.

Cf. A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564 (for base 10).

Sequence in context: A263433 A147806 A064574 * A260295 A024683 A244017

Adjacent sequences:  A059012 A059013 A059014 * A059016 A059017 A059018

KEYWORD

nonn,easy,nice

AUTHOR

Patrick De Geest, Dec 15 2000

STATUS

approved

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Last modified August 21 17:52 EDT 2017. Contains 290892 sequences.