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. (Formerly M2245)

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

0,3

Comments

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

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.

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).

Links

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.

Formula

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

Example

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 + ...

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; # 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

Mathematica

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 *)

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

Crossrefs

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.

Sequence in context: A265903 A345420 A364885 * A097284 A276684 A105353

Adjacent sequences: A006366 A006367 A006368 * A006370 A006371 A006372

Keyword

nonn,nice,easy

Author

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

Extensions

New name from Jon E. Schoenfield, Jul 28 2015

Status

approved