|
| |
|
|
A007494
|
|
Numbers that are congruent to 0 or 2 mod 3.
|
|
61
|
|
|
|
0, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
The map n -> a(n) (where a(n) = 3n/2 if n even or (3n+1)/2 if n odd) was studied by Mahler, in connection with "Z-numbers" and later by Flatto. One question was whether, iterating from an initial integer, one eventually encountered an iterate = 1 (mod 4). - Jeff Lagarias, Sep 23 2002
Partial sums of 0,2,1,2,1,2,1,2,1,... . - Paul Barry, Aug 18 2007
A145389(a(n)) <> 1. - Reinhard Zumkeller, Oct 10 2008
a(n) = A002943(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = numbers k such that antiharmonic mean of the first k positive integers is not integer. A169609(a(n-1)) = 3. See A146535 and A169609. Complement of A016777. - Jaroslav Krizek, May 28 2010
Range of A173732. - Reinhard Zumkeller, Apr 29 2012
Number of partitions of 6n into two odd parts. - Wesley Ivan Hurt, Nov 15 2014
Numbers m such that 3 divides A000217(m). - Bruno Berselli, Aug 04 2017
|
|
|
REFERENCES
|
L. Flatto, Z-numbers and beta-transformations, in Symbolic dynamics and its applications (New Haven, CT, 1991), 181-201, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1002
A. Mader, The Use of Experimental Mathematics in the Classroom, in Interesting Mathematical Problems in Sciences and Everyday Life - 2011.
K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (1968), pp. 313-321.
P. Sabinin and M. G. Stone, Transforming n-gons by Folding the Plane, Amer. Math. Monthly 102, 620-627, 1995.
Eric Weisstein's World of Mathematics, Folding
R. G. Wilson, V, Notes with attachment
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
|
|
|
FORMULA
|
a(n) = 3*n/2 if n even, otherwise (3*n+1)/2.
If u(1)=0, u(n) = n + floor(u(n-1)/3), then a(n-1) = u(n). - Benoit Cloitre, Nov 26 2002
G.f.: x*(x+2)/((1-x)^2*(1+x)). - Ralf Stephan, Apr 13 2002
a(n) = 3*floor(n/2) + 2*(n mod 2) = A032766(n) + A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = (6*n+1)/4 - (-1)^n/4; a(n) = Sum_{k=0..n-1} (1 + (-1)^(k/2)*cos(k*Pi/2)). - Paul Barry, Aug 18 2007
a(n) = 3*n - a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{k>=0} A030308(n,k)*A042950(k). - Philippe Deléham, Oct 17 2011
a(n) = n + ceiling(n/2). - Arkadiusz Wesolowski, Sep 18 2012
a(n) = 2n - floor(n/2) = floor((3n+1)/2) = n + (n + (n mod 2))/2. - Wesley Ivan Hurt, Oct 19 2013
a(n) = A000217(n+1) - A099392(n+1). - Bui Quang Tuan, Mar 27 2015
a(n) = n + floor(n/2) + (n mod 2). - Bruno Berselli, Apr 04 2016
a(n) = Sum_{i=1..n} numerator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k)+(-1)^(k-i). - Wesley Ivan Hurt, Sep 20 2017
|
|
|
MAPLE
|
a[0]:=0:a[1]:=2:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..71); # Zerinvary Lajos, Mar 16 2008
A007494:=n->floor((3*n+1)/2); seq(A007494(k), k=0..100); # Wesley Ivan Hurt, Sep 27 2013
|
|
|
MATHEMATICA
|
Flatten[{#, #+2}&/@(3Range[0, 40])] (* Harvey P. Dale, May 15 2011 *)
Table[2n - Floor[n/2], {n, 0, 100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
|
|
|
PROG
|
(PARI) a(n)=n+(n+1)>>1 \\ Charles R Greathouse IV, Jul 25 2011
(MAGMA) [(6*n+1)/4-(-1)^n/4: n in [0..80]]; // Vincenzo Librandi, Aug 20 2011
(Haskell)
a007494 = flip div 2 . (+ 1) . (* 3) -- Reinhard Zumkeller, Dec 12 2014
|
|
|
CROSSREFS
|
Cf. A000217, A001651, A032766, A035361, A063574, A132462.
Complement of A016777.
Range of A002517.
Cf. A274406. [Bruno Berselli, Jun 26 2016]
Sequence in context: A267528 A294732 A045506 * A258575 A052490 A248562
Adjacent sequences: A007491 A007492 A007493 * A007495 A007496 A007497
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)
|
|
|
STATUS
|
approved
|
| |
|
|
