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A006369
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Nearest integer to 4n/3 unless that is an integer, when 2n/3.
(Formerly M2245)
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14
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0, 1, 3, 2, 5, 7, 4, 9, 11, 6, 13, 15, 8, 17, 19, 10, 21, 23, 12, 25, 27, 14, 29, 31, 16, 33, 35, 18, 37, 39, 20, 41, 43, 22, 45, 47, 24, 49, 51, 26, 53, 55, 28, 57, 59, 30, 61, 63, 32, 65, 67, 34, 69, 71, 36, 73, 75, 38, 77, 79, 40, 81, 83, 42, 85, 87, 44, 89, 91, 46, 93, 95
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| This function was studied by Lothar Collatz in 1932.
Fibonacci numbers lodumo 2 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 26 2009]
a(n)=A006368(n)+A168223(n); A168222(n)=a(a(n)); A168221(a(n))=A006368(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2009]
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REFERENCES
| R. K. Guy, Unsolved Problems in Number Theory, E17.
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 579-581.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 31 (g(n)) and page 270 (f(n)).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| R. Zumkeller, Table of n, a(n) for n = 0..10000 [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 20 2009]
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences that are permutations of the natural numbers
Index entries for two-way infinite sequences
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
| G.f.: x(1+3x+2x^2+3x^3+x^4)/(1-x^3)^2. a(3n)=2n, a(3n+1)=4n+1, a(3n-1)=4n-1, a(-n)=-a(n). - Michael Somos, Oct 05 2003
The map is: n -> if n mod 3 = 0 then 2*n/3 elif n mod 3 = 1 then (4*n-1)/3 else (4*n+1)/3.
a(n) = (2 - ((2*n + 1) mod 3) mod 2) * floor((2*n + 1)/3) + (2*n + 1) mod 3 - 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 23 2005
a(n)=lod_2(F(n)). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 26 2009]
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MAPLE
| A006369 := proc(n) if n mod 3 = 0 then 2*n/3 else round(4*n/3); fi; end;
f:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end; [From N. J. A. Sloane, Feb 04 2011]
A006369:=(1+z**2)*(z**2+3*z+1)/(z-1)**2/(z**2+z+1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[If[Divisible[n, 3], (2n)/3, Floor[(4n)/3+1/2]], {n, 0, 80}] (* From Harvey P. Dale, Nov 03 2011 *)
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PROG
| (PARI) a(n)=if(n%3, round(4*n/3), 2*n/3) - Michael Somos, Oct 05 2003
(Haskell)
a006369 n | m > 0 = round (4 * fromIntegral n / 3)
| otherwise = 2 * n' where (n', m) = divMod n 3
-- Reinhard Zumkeller, Dec 31 2011
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CROSSREFS
| Inverse mapping to A006368. Cf. A028397, A069196.
Trajectories: A028394, A028396, A094328, A094329, A185589, A185590.
Sequence in context: A128224 A125026 A130295 * A097284 A105353 A115966
Adjacent sequences: A006366 A006367 A006368 * A006370 A006371 A006372
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and J. H. Conway (conway(AT)math.princeton.edu)
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