A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.
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
Offset: 0
Examples
9 is odd so a(9) = round(3*9/4) = round(7-1/4) = 7.
References
- 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).
Links
- 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).
Crossrefs
Inverse mapping to A006369.
Programs
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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 -
Magma
[n mod 2 eq 1 select Round(3*n/4) else 3*n/2: n in [0..80]]; // G. C. Greubel, Jan 03 2024
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Maple
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
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 *) -
PARI
a(n)=(3*n+n%2)\(2+n%2*2)
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PARI
a(n)=if(n%2,round(3*n/4),3*n/2)
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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
Formula
If n even, then a(n) = 3*n/2, otherwise, a(n) = round(3*n/4).
G.f.: x*(1+3*x+x^2+3*x^3+x^4)/((1-x^2)*(1-x^4)). - Michael Somos, Jul 23 2002
a(n) = -a(-n).
From Reinhard Zumkeller, Nov 20 2009: (Start)
A168221(n) = a(a(n)).
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
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). (End)
Extensions
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
Comments