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Search: id:A005428
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| A005428 |
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a(0) = 1, state(0) = 2; for n >= 1, if a(n-1) is even then a(n) = 3*a(n-1)/2 and state(n) = state(n-1); if a(n-1) is odd and state(n-1) = 1 then a(n) = ceiling( 3*a(n-1)/2) and state(n) = 3 - state(n-1) and if a(n-1) is odd and state(n-1) = 2 then a(n) = floor( 3*a(n-1)/2) and state(n) = 3 - state(n-1).
(Formerly M0572)
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+0
10
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| 1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 70, 105, 158, 237, 355, 533, 799, 1199, 1798, 2697, 4046, 6069, 9103, 13655, 20482, 30723, 46085, 69127, 103691, 155536, 233304, 349956, 524934, 787401, 1181102, 1771653, 2657479, 3986219, 5979328, 8968992, 13453488, 20180232, 30270348, 45405522, 68108283, 102162425, 153243637, 229865456, 344798184
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Arises from a version of the Josephus problem: sequence gives set of n where, if you start with n people and every 3rd person drops out, either it is person #1 or #2 who is left at the end. A081614 and A081615 give the subsequences where it is person #1 (respectively #2) who is left.
The state changes just when a(n) is odd: it therefore indicates whether the sum of a(0) to a(n) is odd (1 means no, 2 means yes).
The sum a(0) to a(n) is never divisible by 3 (for n >= 0); it is 1 mod 3 precisely when the sum a(0) to a(n-1) is odd and thus indicates the state at the previous step. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 14 2008
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REFERENCES
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K. Burde, Das Problem der Abzaehlreime und Zahlentwicklungen mit gebrochenen Basen, J. Number Theory 26 (1987), no. 2, 192-209.
F. Schuh, The Master Book of Mathematical Recreations. Dover, NY, 1968, page, 374, Table 18, union of columns 1 and 2 (which are A081614 and A081615).
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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T. D. Noe, Table of n, a(n) for n=0..500
L. Halbeisen and N. Hungerbuehler, The Josephus Problem
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem
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FORMULA
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a(0) = 1; a(n) = ceiling[(1 + sum{ a(0) to a(n-1) }) / 2]. - Don Reble (djr(AT)nk.ca), Apr 23, 2003
a(1) = 1, s(1) = 2, and for n>1, a(n) = floor( (3*a(n-1) + s(n-1) - 1) / 2 ), s(n) = (s(n-1) + a(n)) mod 3. [From Max Alekseyev (maxale(AT)gmail.com), Mar 28 2009]
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EXAMPLE
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n........0...1...2...3...4...5...6...7...8...9..10..11..12..13..14.
state=1......1...........4...6...9..........31.....70..105.........
state=2..1.......2...3..............14..21......47.........158..237
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MATHEMATICA
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f[s_] := Append[s, Ceiling[(1 + Plus @@ s)/2]]; Nest[f, {1}, 40] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jul 07 2006)
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PROGRAM
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(PARI) { a=1; s=2; for(k=1, 50, print1(a, ", "); a=(3*a+s-1)\2; s=(s+a)%3; ) } [From Max Alekseyev (maxale(AT)gmail.com), Mar 28 2009]
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CROSSREFS
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Cf. A005427, A073941, A082416. Union of A081614 and A081615.
First differences of D_3(n) (A061419) in the terminology of Odlyzko and Wilf. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 23 2002
Same as log2(A082125(n+3)). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 16 2002
Apart from initial terms, same as A073941, which has further information.
Partial sums are in A061419, A061418, A006999.
Sequence in context: A039884 A078620 A073941 this_sequence A143951 A058355 A099558
Adjacent sequences: A005425 A005426 A005427 this_sequence A005429 A005430 A005431
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com) and Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
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More terms from Hans Havermann (pxp(AT)rogers.com), Apr 23, 2003
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