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A014682
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The Collatz or 3x+1 function: a(n) = n/2 if n is even, otherwise (3n+1)/2.
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22
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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; internal format)
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OFFSET
| 0,2
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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 '2,5,8,11,... ("add 3")' and the nonnegative integers.
a(n) = log to the base 2 of A076936(n). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 19 2002
Partial sums are A093353. - Paul Barry (pbarry(AT)wit.ie), Mar 31 2008
Absolute first differences are essentially in A014681 and A103889. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 05 2008
a(n)= 5/4 + (1/2)*((-1)^n)*n + (3/4)*(-1)^n + n - Alexander R. Povolotsky (pevnev(AT)juno.com), Apr 05 2008
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REFERENCES
| J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..1000
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
| G.f.: x(2+x+x^2)/(1-x^2)^2; a(n)=(4n+1)/4-(2n+1)(-1)^n/4 [offset 0]. - Paul Barry (pbarry(AT)wit.ie), Mar 31 2008
a(n) = -a(n-1) + a(n-2) + a(n-3) + 4 - John W. Layman (layman(AT)math.vt.edu)
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 (benoit7848c(AT)orange.fr), May 12 2002
O.g.f.: x*(2+x+x^2)/[(-1+x)^2*(1+x)^2]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 05 2008
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MAPLE
| T:=proc(n) if n mod 2 = 0 then n/2 else (3*n+1)/2; fi; end; (From 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 (mathar(AT)strw.leidenuniv.nl), Mar 20 2007
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MATHEMATICA
| Collatz[n_?OddQ] := (3n + 1)/2; Collatz[n_?EvenQ] := n/2; Table[Collatz[n], {n, 0, 79}] (* From Alonso del Arte, Apr 21 2011 *)
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CROSSREFS
| Cf. A076936, A076938, A016116, A126241, A060412, A060413.
Cf. A006370.
Sequence in context: A185727 A070951 A076937 * A111361 A167160 A205377
Adjacent sequences: A014679 A014680 A014681 * A014683 A014684 A014685
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KEYWORD
| nonn,easy
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AUTHOR
| Mohammad K. Azarian (ma3(AT)evansville.edu)
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 26 2008, at the suggestion of Artur Jasinski (grafix(AT)csl.pl)
Edited by N. J. A. Sloane, Jan 31 2011
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