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A061168 Partial sums of A000523. 11
0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 28, 31, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 103, 108, 113, 118, 123, 128, 133, 138, 143, 148, 153, 158, 163, 168, 173, 178, 183, 188, 193, 198, 203, 208, 213, 218, 223, 228, 233, 238, 243, 248 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Given a term b>0 of the sequence and its left hand neighbor c, the corresponding unique sequence index n with property a(n)=b can be determined by n(b)=e+(b-d*(e+1)+2*(e-1))/d, where d=b-c and e=2^d. - Hieronymus Fischer, Dec 05 2006

a(n) gives index of start of binary expansion of n in the binary Champernowne sequence A076478. - N. J. A. Sloane, Dec 14 2017

REFERENCES

D. E. Knuth, Fundamental Algorithms, Addison-Wesley, 1973, Section 1.2.4, ex. 42(b).

LINKS

Harry J. Smith, Table of n, a(n) for n=1..1000

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197, ex. 27.

Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012.

Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75.

M. Griffiths, More sums involving the floor function, Math. Gaz., 86 (2002), 285-287.

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.

FORMULA

a(n) = A001855(n+1) - n.

a(n) = Sum_{k=1..n} floor(log_2(k)) = (n+1)*floor(log_2(n)) - 2*(2^floor(log_2(n)) - 1). - Diego Torres (torresvillarroel(AT)hotmail.com), Oct 29 2002

G.f.: 1/(1-x)^2 * Sum(k>=1, x^2^k). - Ralf Stephan, Apr 13 2002

a(n) = A123753(n) - 2*n - 1. - Peter Luschny, Nov 30 2017

MAPLE

[seq(add(floor(log_2(k)), k=1..j), j=1..100)]

MATHEMATICA

Accumulate[Floor[Log[2, Range[110]]]] (* Harvey P. Dale, Jul 16 2012 *)

a[n_] := (n+1) IntegerLength[n+1, 2] - 2^IntegerLength[n+1, 2] - n + 1;

Table[a[n], {n, 1, 61}] (* Peter Luschny, Dec 02 2017 *)

PROG

(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)+a(n/2-1)+n-1, 2*a((n-1)/2)+n-1)) /* _Ralf Stephan */

(PARI) a(n)=local(k); if(n<1, 0, k=length(binary(n))-1; (n+1)*k-2*(2^k-1))

(PARI) { for (n=1, 1000, k=length(binary(n))-1; write("b061168.txt", n, " ", (n + 1)*k - 2*(2^k - 1)) ) } \\ Harry J. Smith, Jul 18 2009

(Haskell)

import Data.List (transpose)

a061168 n = a061168_list !! n

a061168_list = zipWith (+) [0..] (zipWith (+) hs $ tail hs) where

   hs = concat $ transpose [a001855_list, a001855_list]

-- Reinhard Zumkeller, Jun 03 2013

(Python)

def A061168(n):

    s, i, z = -n , n, 1

    while 0 <= i: s += i; i -= z; z += z

    return s

print([A061168(n) for n in range(1, 62)]) # Peter Luschny, Nov 30 2017

CROSSREFS

Cf. A001855, A123753, A076478.

Sequence in context: A053044 A129011 A130174 * A130798 A165453 A282168

Adjacent sequences:  A061165 A061166 A061167 * A061169 A061170 A061171

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Apr 19 2001

STATUS

approved

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Last modified February 23 15:33 EST 2018. Contains 299581 sequences. (Running on oeis4.)