This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A059015 Total number of 0's in binary expansions of 0, ..., n. 34
 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) Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016. Hsien-Kuei Hwang, S. Janson, T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585. Jeffrey C. Lagarias, The Takagi function and its properties, arXiv:1112.4205 [math.CA], 2011-2012. Jeffrey C. Lagarias, 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. R. Stephan, Some divide-and-conquer sequences ... R. Stephan, Table of generating functions 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 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 - x) + (1/(1 - x)^2)*Sum_{j>=0} x^(2*2^j)/(1 + x^(2^j)); corrected by Ilya Gutkovskiy, Mar 28 2018 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))<

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 14 05:08 EDT 2019. Contains 327995 sequences. (Running on oeis4.)