A168221 -id:A168221 - cp's OEIS frontend

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

N. J. A. Sloane and J. H. Conway

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)
Starting at n = 1, the sequence equals the smallest unused positive number such that a(n)-a(n-1) does not appear as a term in the current sequence. - Scott R. Shannon, Dec 20 2023

			9 is odd so a(9) = round(3*9/4) = round(7-1/4) = 7.

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).

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).

Inverse mapping to A006369.
Cf. A047529, A178414, A178415, A265667.
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.

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

[n mod 2 eq 1 select Round(3*n/4) else 3*n/2: n in [0..80]]; // G. C. Greubel, Jan 03 2024

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.]

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)
```
PARI
```
a(n)=if(n%2,round(3*n/4),3*n/2)
```

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

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)
a(n) = A006369(n) - A168223(n).
A168221(n) = a(a(n)).
A168222(a(n)) = A006369(n). (End)
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)

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

A006369 a(n) = 2n/3 for n divisible by 3, otherwise a(n) = round(4n/3). Or, equivalently, a(3n-2) = 4n-3, a(3n-1) = 4n-1, a(3n) = 2n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane and J. H. Conway

Original name was: Nearest integer to 4n/3 unless that is an integer, when 2n/3.
This function was studied by Lothar Collatz in 1932.
Fibonacci numbers lodumo_2. - Philippe Deléham, Apr 26 2009
a(n) = A006368(n) + A168223(n); A168222(n) = a(a(n)); A168221(a(n)) = A006368(n). - Reinhard Zumkeller, Nov 20 2009
The permutation P given in A265667 is P(n) = a(n-1) + 1, for n >= 0, with a(-1) = -1. Observed by Kevin Ryde. - Wolfdieter Lang, Sep 22 2021

			G.f. = x + 3*x^2 + 2*x^3 + 5*x^4 + 7*x^5 + 4*x^6 + 9*x^7 + 11*x^8 + 6*x^9 + ...

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.
K. Knopp, Infinite Sequences and Series, Dover Publications, NY, 1958, p. 77.
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).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.
M. Klamkin, Proposer, An infinite permutation, Problem 63-13, SIAM Review, Vol. 8:2 (1966), 234-236.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
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 two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Index entries for sequences related to 3x+1 (or Collatz) problem.
Index entries for sequences that are permutations of the natural numbers.

Inverse mapping to A006368.
Cf. A028397, A069196, A265667.
Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589.

a006369 n | m > 0     = round (4 * fromIntegral n / 3)
          | otherwise = 2 * n' where (n',m) = divMod n 3
-- Reinhard Zumkeller, Dec 31 2011

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; # N. J. A. Sloane, Feb 04 2011
A006369:=(1+z**2)*(z**2+3*z+1)/(z-1)**2/(z**2+z+1)**2; # Simon Plouffe, in his 1992 dissertation

Table[If[Divisible[n,3],(2n)/3,Floor[(4n)/3+1/2]],{n,0,80}] (* Harvey P. Dale, Nov 03 2011 *)
Table[n + Floor[(n + 1)/3] (-1)^Mod[n + 1, 3], {n, 0, 80}] (* Bruno Berselli, Dec 10 2015 *)

{a(n) = if( n%3, round(4*n / 3), 2*n / 3)}; /* Michael Somos, Oct 05 2003 */

From Michael Somos, Oct 05 2003: (Start)
G.f.: x * (1 + 3*x + 2*x^2 + 3*x^3 + x^4) / (1 - x^3)^2.
a(3*n) = 2*n, a(3*n + 1) = 4*n + 1, a(3*n - 1) = 4*n - 1, a(n) = -a(-n) for all n in Z. (End)
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, Jan 23 2005
a(n) = lod_2(F(n)). - Philippe Deléham, Apr 26 2009
0 = 21 + a(n)*(18 + 4*a(n) - a(n+1) - 7*a(n+2)) + a(n+1)*(-a(n+2)) + a(n+2)*(-18 + 4*a(n+2)) for all n in Z. - Michael Somos, Aug 24 2014
a(n) = n + floor((n+1)/3)*(-1)^((n+1) mod 3). - Bruno Berselli, Dec 10 2015
a(n) = 2*a(n-3) - a(n-6) for n >= 6. - Werner Schulte, Mar 16 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(sqrt(2)+2)/sqrt(2) + (1-sqrt(2)/2)*log(2)/2. - Amiram Eldar, Sep 29 2022

New name from Jon E. Schoenfield, Jul 28 2015

A168222 a(n) = A006369(A006369(n)).

Original entry on oeis.org

0, 1, 2, 3, 7, 9, 5, 6, 15, 4, 17, 10, 11, 23, 25, 13, 14, 31, 8, 33, 18, 19, 39, 41, 21, 22, 47, 12, 49, 26, 27, 55, 57, 29, 30, 63, 16, 65, 34, 35, 71, 73, 37, 38, 79, 20, 81, 42, 43, 87, 89, 45, 46, 95, 24, 97, 50, 51, 103, 105, 53, 54, 111, 28, 113, 58, 59, 119, 121, 61, 62

Offset: 0

Reinhard Zumkeller, Nov 20 2009

nonn

Inverse integer permutation to A168221;

a(A006368(n)) = A006369(n).

Table[Nest[If[Divisible[#,3],2#/3,Round[4#/3]]&,n,2],{n,0,100}] (* Paolo Xausa, Dec 15 2023 *)

Conjectures from Colin Barker, Aug 15 2019: (Start)
G.f.: x*(1 + 2*x + 3*x^2 + 7*x^3 + 9*x^4 + 5*x^5 + 6*x^6 + 15*x^7 + 4*x^8 + 15*x^9 + 6*x^10 + 5*x^11 + 9*x^12 + 7*x^13 + 3*x^14 + 2*x^15 + x^16) / ((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2).
a(n) = 2*a(n-9) - a(n-18) for n>17.
(End)

A368179 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006368 map.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 6, 3, 3, 1, 0, 6, 4, 9, 2, 2, 1, 0, 7, 9, 6, 7, 3, 3, 1, 0, 8, 5, 7, 9, 5, 2, 2, 1, 0, 9, 12, 4, 5, 7, 4, 3, 3, 1, 0, 10, 7, 18, 6, 4, 5, 6, 2, 2, 1, 0, 11, 15, 5, 27, 9, 6, 4, 9, 3, 3, 1, 0, 12, 8, 11, 4, 20, 7, 9, 6, 7, 2, 2, 1, 0

Offset: 0

Paolo Xausa, Dec 15 2023

			Array begins:
  [ 0]   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ... = A000004
  [ 1]   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ... = A000012
  [ 2]   2,  3,  2,  3,  2,  3,  2,  3,  2,  3,  2, ... = A010693
  [ 3]   3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3, ... = A176059
  [ 4]   4,  6,  9,  7,  5,  4,  6,  9,  7,  5,  4, ... = A180853
  [ 5]   5,  4,  6,  9,  7,  5,  4,  6,  9,  7,  5, ... = A180853 (shifted)
  [ 6]   6,  9,  7,  5,  4,  6,  9,  7,  5,  4,  6, ... = A180853 (shifted)
  [ 7]   7,  5,  4,  6,  9,  7,  5,  4,  6,  9,  7, ... = A180853 (shifted)
  [ 8]   8, 12, 18, 27, 20, 30, 45, 34, 51, 38, 57, ... = A028393
  [ 9]   9,  7,  5,  4,  6,  9,  7,  5,  4,  6,  9, ... = A180853 (shifted)
  [10]  10, 15, 11,  8, 12, 18, 27, 20, 30, 45, 34, ... = A180864 (shifted)
  ...    |   |   |
      A001477|A168221
             |
          A006368

Paolo Xausa, Table of n, a(n) for n = 0..11324 (antidiagonals 1..150 of the array, flattened).
Index entries for sequences related to 3x+1 (or Collatz) problem

Rows: A000004, A000012, A010693,  A028393, A028395, A094332, A176059, A180853, A180864, A182205, A217218, A223086, A223087, A223088.
Columns: A001477, A006368, A168221.
Main diagonal: A368180.

A006368[n_]:=If[OddQ[n],Floor[(3n+2)/4],3n/2];
A368179list[dmax_]:=With[{a=Reverse[Table[NestList[A006368,n-1,dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368179list[15] (* Generates 15 antidiagonals *)

cp's OEIS Frontend

A006368 The "amusical permutation" of the nonnegative numbers: a(2n)=3n, a(4n+1)=3n+1, a(4n-1)=3n-1.

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A006369 a(n) = 2*n/3 for n divisible by 3, otherwise a(n) = round(4*n/3). Or, equivalently, a(3*n-2) = 4*n-3, a(3*n-1) = 4*n-1, a(3*n) = 2*n.

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A168222 a(n) = A006369(A006369(n)).

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A368179 Square array read by ascending antidiagonals: row n is the trajectory of n under the A006368 map.

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Author

Keywords

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Links

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Programs

Mathematica

A006369 a(n) = 2n/3 for n divisible by 3, otherwise a(n) = round(4n/3). Or, equivalently, a(3n-2) = 4n-3, a(3n-1) = 4n-1, a(3n) = 2n.