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A060973 a(2*n+1) = a(n+1)+a(n), a(2*n) = 2*a(n), with a(1)=0 and a(2)=1. 5
0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..1000

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16.

H.-K. Hwang, S. Janson, and 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.

H.-K. Hwang, S. Janson, and 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 Shallit, Intertwining of Complementary Thue-Morse Factors, arXiv:2203.02917 [cs.FL], 2022.

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

Ralf Stephan, Table of generating functions

FORMULA

a(n) = n-A006165(n) = A006165(n)-A053646(n) = (n-A053646(n))/2 [for n>1 ]. If n = 2*2^m+k with 0< = k< = 2^m, then a(n) = 2^m; if n = 3*2^m+k with 0< = k< = 2^m, then a(n) = 2^m+k.

G.f. -x/(1-x) + x/(1-x)^2 * (1 + sum(k>=0, t^2(t-1), t=x^2^k)). - Ralf Stephan, Sep 12 2003

EXAMPLE

a(6)=2*a(3)=2*1=2. a(7)=a(3)+a(4)=1+2=3.

MAPLE

A060973 := proc(n)

    option remember;

    if n <= 2 then

        return n-1;

    fi;

    if n mod 2 = 0 then

        2*procname(n/2)

    else

        procname((n-1)/2)+procname((n+1)/2);

    fi;

end proc:

MATHEMATICA

nn = 77; Array[Set[a[#], # - 1] &, 2]; Do[Set[a[i], If[EvenQ[i], 2 a[i/2], a[# + 1] + a[#] &[(i - 1)/2]]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 22 2022 *)

PROG

(Python)

from functools import lru_cache

@lru_cache(maxsize=None)

def A060973(n): return n-1 if n <= 2 else A060973(n//2) + A060973((n+1)//2) # Chai Wah Wu, Mar 08 2022

CROSSREFS

Cf. A006165, A053646.

Sequence in context: A228482 A091822 A173022 * A352228 A097915 A255072

Adjacent sequences:  A060970 A060971 A060972 * A060974 A060975 A060976

KEYWORD

nonn

AUTHOR

Henry Bottomley, May 09 2001

STATUS

approved

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Last modified July 3 04:45 EDT 2022. Contains 355030 sequences. (Running on oeis4.)