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A014682 The Collatz or 3x+1 function: a(n) = n/2 if n is even, otherwise (3n+1)/2. 63
0, 2, 1, 5, 2, 8, 3, 11, 4, 14, 5, 17, 6, 20, 7, 23, 8, 26, 9, 29, 10, 32, 11, 35, 12, 38, 13, 41, 14, 44, 15, 47, 16, 50, 17, 53, 18, 56, 19, 59, 20, 62, 21, 65, 22, 68, 23, 71, 24, 74, 25, 77, 26, 80, 27, 83, 28, 86, 29, 89, 30, 92, 31, 95, 32, 98, 33, 101, 34, 104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the function usually denoted by T(n) in the literature on the 3x+1 problem. See A006370 for further references and links.

Intertwining of sequence A016789 '2,5,8,11,... ("add 3")' and the nonnegative integers.

a(n) = log to the base 2 of A076936(n). - Amarnath Murthy, Oct 19 2002

Partial sums are A093353. - Paul Barry, Mar 31 2008

Absolute first differences are essentially in A014681 and A103889. - R. J. Mathar, Apr 05 2008

The Monks article claims to "determine the structure of the groups generated by the maps x maps to x/2 and x maps to (3x+1)/2 modulo b for b relatively prime to 6, and study the action of these groups on the directed graph associated to the 3x+1 dynamical system." - Jonathan Vos Post, Apr 18 2012

Only terms of A016789 occur twice, at positions given by sequences A005408 (odd numbers) and A016957 (6n+4): (1,4), (3,10), (5,16), (7,22), ... - Antti Karttunen, Jul 28 2017

REFERENCES

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.

LINKS

N. J. A. Sloane (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..100000

C. J. Everett, Iteration of the number-theoretic function f(2n) = n, f(2n + 1) = 3n + 2, Advances in Mathematics, Volume 25, Issue 1, July 1977, Pages 42-45.

Keenan Monks, Kenneth G. Monks, Kenneth M. Monks, Maria Monks, Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+1 graph, arXiv:1204.3904v1 [math.DS], Apr 17 2012.

R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

Index entries for sequences related to 3x+1 (or Collatz) problem

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

FORMULA

From Paul Barry, Mar 31 2008: (Start)

G.f.: x(2 + x + x^2)/(1-x^2)^2.

a(n) = (4n+1)/4-(2n+1)(-1)^n/4 [offset 0]. (End)

a(n) = -a(n-1) + a(n-2) + a(n-3) + 4. - John W. Layman

For n > 1 this is the image of n under the modified "3x+1" map (cf. A006370): n -> n/2 if n is even, n -> (3n+1)/2 if n is odd. - Benoit Cloitre, May 12 2002

O.g.f.: x*(2+x+x^2)/[(-1+x)^2*(1+x)^2]. - R. J. Mathar, Apr 05 2008

a(n) = 5/4 + (1/2)*((-1)^n)*n + (3/4)*(-1)^n + n. - Alexander R. Povolotsky, Apr 05 2008

a(n) = Sum_{i=-n..2*n} i*(-1)^i. [Bruno Berselli, Dec 14 2015]

a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k) + (-1)^k. - Wesley Ivan Hurt, Sep 20 2017

EXAMPLE

a(3) = -3*(-1)-2*1-1*(-1)-0*1+1*(-1)+2*1+3*(-1)+4*1+5*(-1)+6*1 = 5. [Bruno Berselli, Dec 14 2015]

MAPLE

T:=proc(n) if n mod 2 = 0 then n/2 else (3*n+1)/2; fi; end; # N. J. A. Sloane, Jan 31 2011

A076936 := proc(n) option remember ; local apr, ifr, me, i, a ; if n <=2 then n^2 ; else apr := mul(A076936(i), i=1..n-1) ; ifr := ifactors(apr)[2] ; me := -1 ; for i from 1 to nops(ifr) do me := max(me, op(2, op(i, ifr))) ; od ; me := me+ n-(me mod n) ; a := 1 ; for i from 1 to nops(ifr) do a := a*op(1, op(i, ifr))^(me-op(2, op(i, ifr))) ; od ; if a = A076936(n-1) then me := me+n ; a := 1 ; for i from 1 to nops(ifr) do a := a*op(1, op(i, ifr))^(me-op(2, op(i, ifr))) ; od ; fi ; RETURN(a) ; fi ; end: A014682 := proc(n) log[2](A076936(n)) ; end: for n from 1 to 85 do printf("%d, ", A014682(n)) ; od ; # R. J. Mathar, Mar 20 2007

MATHEMATICA

Collatz[n_?OddQ] := (3n + 1)/2; Collatz[n_?EvenQ] := n/2; Table[Collatz[n], {n, 0, 79}] (* Alonso del Arte, Apr 21 2011 *)

LinearRecurrence[{0, 2, 0, -1}, {0, 2, 1, 5}, 70] (* Jean-Fran├žois Alcover, Sep 23 2017 *)

PROG

(Haskell)

a014682 n = if r > 0 then div (3 * n + 1) 2 else n'

            where (n', r) = divMod n 2

-- Reinhard Zumkeller, Oct 03 2014

(PARI) a(n)=if(n%2, 3*n+1, n)/2 \\ Charles R Greathouse IV, Sep 02 2015

(Python)

def a(n): return n/2 if n%2==0 else (3*n + 1)/2

print map(a, xrange(101)) # Indranil Ghosh, Jul 29 2017

CROSSREFS

Cf. A076936, A076938, A139391, A016116, A126241, A060412, A060413, A006370, A070168 (iterations), A005408, A016957, A064455, A153285.

Bisections: A001477, A016789.

Sequence in context: A185727 A070951 A076937 * A111361 A167160 A257971

Adjacent sequences:  A014679 A014680 A014681 * A014683 A014684 A014685

KEYWORD

nonn,easy,changed

AUTHOR

Mohammad K. Azarian

EXTENSIONS

Edited by N. J. A. Sloane, Apr 26 2008, at the suggestion of Artur Jasinski

Edited by N. J. A. Sloane, Jan 31 2011

STATUS

approved

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Last modified September 25 16:31 EDT 2017. Contains 292499 sequences.