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A006165 a(1) = a(2) = 1; thereafter a(2n+1) = a(n+1) + a(n), a(2n) = 2a(n).
(Formerly M0277)
7
1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 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, 32, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n+1) is the second-order survivor of the n-person Josephus problem where every second person is marked until only one remains, who is then eliminated; the process is repeated from the beginning until all but one is eliminated. a(n) is first a power of 2 when n is three times a power of 2. For example, the first appearances of 2, 4, 8 and 16 are at positions 3, 6, 12 and 24, or (3*1),(3*2),(3*4) and (3*8). Eugene McDonnell (eemcd(AT)aol.com), Jan 19 2002, reporting on work of Boyko Bantchev (Bulgaria).

a(n+1)=min(msb(n),1+n-msb(n)/2) for all n (msb = most significant bit, A053644). - Boyko Bantchev (bantchev(AT)math.bas.bg), May 17 2002

Appears to coincide with following sequence: Let n >= 1. Start with a bag B containing n 1's. At each step, replace the two least elements x and y in B with the single element x+y. Repeat until B contains 2 or fewer elements. Let a(n) be the largest element remaining in B at this point. - David W. Wilson, Jul 01 2003

Hsien-Kuei Hwang, S Janson, TH Tsai (2016) show that A078881 is the same sequence, apart from the offset. - N. J. A. Sloane, Nov 26 2017

REFERENCES

J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.

Hsien-Kuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also 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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1024

J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

Dale Gerdemann, Second-Order Josephus Problem (video)

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

R. Stephan, Table of generating functions

FORMULA

For n>=2, if a(n)>=A006257(n), i.e., if msb(n)>n-a(n)/2, then a(n+1)=a(n)+1, otherwise a(n+1)=a(n). - Henry Bottomley, Jan 21 2002

a(1)=1, a(n)=n-a(n-a(a(n-1))). - Benoit Cloitre, Nov 08 2002

For k>0, 0<=i<=2^k-1, a(2^k+i)=2^(k-1)+i; for 2^k-2^(k-2)<=x<=2^k a(x)=2^(k-1); (also a(m*2^k)=a(m)*2^k for m>=2). - Benoit Cloitre, Dec 16 2002

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

a(n) = A005942(n+1)/2 - n = n - A060973(n) = 2n - A007378(n). - Ralf Stephan, Sep 13 2003

a(n) = A080776(n-1) + A060937(n). - Ralf Stephan

MATHEMATICA

t = {1, 1}; Do[If[OddQ[n], AppendTo[t, t[[Floor[n/2]]] + t[[Ceiling[n/2]]]], AppendTo[t, 2*t[[n/2]]]], {n, 3, 128}] (* T. D. Noe, May 25 2011 *)

CROSSREFS

Cf. A066997, A078881.

Sequence in context: A076502 A076897 A066997 * A078881 A131807 A104351

Adjacent sequences:  A006162 A006163 A006164 * A006166 A006167 A006168

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2002

STATUS

approved

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Last modified February 23 23:15 EST 2018. Contains 299595 sequences. (Running on oeis4.)