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A006165
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a(1) = a(2) = 1; thereafter a(2n+1) = a(n+1) + a(n), a(2n) = 2a(n).
(Formerly M0277)
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17
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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
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OFFSET
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1,3
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COMMENTS
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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).
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
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REFERENCES
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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).
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LINKS
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FORMULA
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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(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
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)) where t = x^2^k. - Ralf Stephan, Sep 12 2003
For k a positive integer, define the k-th iterated sequence a^(k) of a by a^(1)(n) = a(n) and setting a^(k)(n) = a^(k-1)(a(n)) for k >= 2. For example, a^(2)(n) = a(a(n)) and a^(3)(n) = a(a(a(n))).
Conjectures: for n >= 2 there holds
(i) a(n) + a(n - a(n - a(n - a(n - a(n))))) = n;
(ii) a(n - a(n - a(n - a(n)))) = a(n - a(n - a(n - a(n - a(n - a(n))))));
(iii) a^2(n) = a(n - a(n - a(n - a(n))));
(iv) n - a(n) = a(n - a^(2)(n));
(v) a(n - a(n)) = a^(2)(n - a^(2)(n - a^(2)(n - a^(2)(n))));
(vi) for k >= 2, a^(k)(n - a^(k)(n)) = a^(k)(n - a^(k)(n - a^(k)(n - a^(k)(n)))).
(vii) for k >= 1, the sequence {n - a^(k)(n) : n >= 1} has first differences either 0 or 1. We conjecture that the repeated values of the sequence are of the form (2^k - 1)*2^m. The number of repeated values appears to always be 2, 3, 5, 9, 17, 35, ..., independent of k, conjecturally A000051. Two examples are given below.
A similar property may hold for the sequences {n - A060973^(k)(n) : n >= 2^(k-1)}, k = 1,2,3,.... (End)
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EXAMPLE
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1) The sequence {n - a(a(n)) : n >= 1} begins [0, 1, 2, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 49, ...] has the repeated values 3 (twice), 6 (three times), 12 (five times), 24 (nine times), 48 (seventeen times) ..., conjecturally of the form 3*2^m
2) The sequence {n - a(a(a(n))) : n >= 1} begins [0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28, 28, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 56, 56, 56, 56, 56, 56, 56, 56, 57, ...] has the repeated values 7 (twice), 14 (three times), 28 (five times), 56 (nine times) ..., conjecturally of the form 7*2^m. (End)
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MAPLE
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a := proc (n) option remember; if n = 1 then 1 else n - a(n - a(a(n-1))) end if end proc: seq(a(n), n = 1..100); # Peter Bala, Jul 31 2022
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MATHEMATICA
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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 *)
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PROG
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(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
(PARI) a(n) = my(i=logint(n, 2)-1); if(bittest(n, i), 2<<i, n - 1<<i); \\ Kevin Ryde, Aug 06 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jun 12 2002
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STATUS
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approved
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