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A006368
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The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.
(Formerly M2249)
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42
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0, 1, 3, 2, 6, 4, 9, 5, 12, 7, 15, 8, 18, 10, 21, 11, 24, 13, 27, 14, 30, 16, 33, 17, 36, 19, 39, 20, 42, 22, 45, 23, 48, 25, 51, 26, 54, 28, 57, 29, 60, 31, 63, 32, 66, 34, 69, 35, 72, 37, 75, 38, 78, 40, 81, 41, 84, 43, 87, 44, 90, 46, 93, 47, 96, 49, 99, 50, 102, 52, 105, 53
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OFFSET
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0,3
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COMMENTS
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A permutation of the nonnegative integers.
There is a famous open question concerning the closed trajectories under this map - see A217218, A028393, A028394, and Conway (2013).
This is lodumo_3 of A131743. - Philippe Deléham, Oct 24 2011
Multiples of 3 interspersed with numbers other than multiples of 3. - Harvey P. Dale, Dec 16 2011
For n>0: a(2n+1) is the smallest number missing from {a(0),...,a(2n-1)} and a(2n) = a(2n-1) + a(2n+1). - Bob Selcoe, May 24 2017
From Wolfdieter Lang, Sep 21 2021: (Start)
The permutation P of positive natural numbers with P(n) = a(n-1) + 1, for n >= 1, is the inverse of the permutation given in A265667, and it maps the index n of A178414 to the index of A047529: A178414(n) = A047529(P(n)).
Thus each number {1, 3, 7} (mod 8) appears in the first column A178414 of the array A178415 just once. For the formulas see below. (End)
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REFERENCES
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J. H. Conway, Unpredictable iterations, in Proc. Number Theory Conf., Boulder, CO, 1972, pp. 49-52.
R. K. Guy, Unsolved Problems in Number Theory, E17.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 5.
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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R. Zumkeller, Table of n, a(n) for n = 0..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7..
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198. [Introduces the name "amusical permutation".]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
S. Schreiber & N. J. A. Sloane, Correspondence, 1980
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for two-way infinite sequences
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).
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FORMULA
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If n even, then a(n) = 3n/2; otherwise, a(n) = nearest integer to 3n/4.
G.f.: x(1+3x+x^2+3x^3+x^4)/((1-x^2)(1-x^4)). - Michael Somos, Jul 23 2002
a(n) = -a(-n).
a(n) = A006369(n) - A168223(n); A168221(n) = a(a(n)); A168222(a(n)) = A006369(n). - Reinhard Zumkeller, Nov 20 2009
a(n) = a(n-2) + a(n-4) - a(n-6); a(0)=0, a(1)=1, a(2)=3, a(3)=2, a(4)=6, a(5)=4. - Harvey P. Dale, Dec 16 2011
a(n) = ((n+1) mod 2) * (3n/2) + (n mod 2) * round(3n/4). - Wesley Ivan Hurt, Nov 23 2013
From Wolfdieter Lang, Sep 21 2021: (Start)
Formulas for the permutation P(n) = a(n-1) + 1 mentioned above:
P(n) = n + floor(n/2) if n is odd, and n - floor(n/4) if n is even.
P(n) = (3*n-1)/2 if n is odd; P(n) = (3*n+2)/4 if n == 2 (mod 4); and P(n) = 3*n/4 if n == 0 (mod 4).
O.g.f.: x*(1 + 2*x + 3*x^2 + x^3 + 2*x^4)/((1-x^2)*(1-x^4)). (End)
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EXAMPLE
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9 is odd so a(9) = round(3*9/4) = round(7-1/4) = 7.
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MAPLE
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f:=n-> if n mod 2 = 0 then 3*n/2 elif n mod 4 = 1 then (3*n+1)/4 else (3*n-1)/4; fi; # N. J. A. Sloane, Jan 21 2011
A006368:=(1+3*z+z**2+3*z**3+z**4)/(1+z**2)/(z-1)**2/(1+z)**2; # [Conjectured (correctly, except for the offset) by Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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Table[If[EvenQ[n], (3n)/2, Floor[(3n+2)/4]], {n, 0, 80}] (* or *) LinearRecurrence[ {0, 1, 0, 1, 0, -1}, {0, 1, 3, 2, 6, 4}, 80] (* Harvey P. Dale, Dec 16 2011 *)
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PROG
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(PARI) a(n)=(3*n+n%2)\(2+n%2*2)
(PARI) a(n)=if(n%2, round(3*n/4), 3*n/2)
(Haskell)
a006368 n | u' == 0 = 3 * u
| otherwise = 3 * v + (v' + 1) `div` 2
where (u, u') = divMod n 2; (v, v') = divMod n 4
-- Reinhard Zumkeller, Apr 18 2012
(Python)
def a(n): return 0 if n == 0 else 3*n//2 if n%2 == 0 else (3*n+1)//4
print([a(n) for n in range(72)]) # Michael S. Branicky, Aug 12 2021
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CROSSREFS
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Inverse mapping to A006369.
Cf. A028393, A028294, A028397, A180853, A180864, A182205, A217218.
Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.
Cf. A047529, A178414, A178415, A265667.
Sequence in context: A257903 A257877 A257910 * A202845 A202838 A105354
Adjacent sequences: A006365 A006366 A006367 * A006369 A006370 A006371
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane and J. H. Conway
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EXTENSIONS
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Edited by Michael Somos, Jul 23 2002
I replaced the definition with the original definition of Conway and Guy. - N. J. A. Sloane, Oct 03 2012
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STATUS
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approved
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