A007494 Numbers that are congruent to 0 or 2 mod 3.

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

Offset: 0

Author

Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)

Keywords

• nonn
• easy

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

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

Maximal length of a snake like polyomino that fits in a 2 X n rectangle. - Alain Goupil, Feb 12 2020

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.
• Attila Máder, The Use of Experimental Mathematics in the Classroom, in Interesting Mathematical Problems in Sciences and Everyday Life - 2011.
• Kurt Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc., Vol. 8 (1968), pp. 313-321.
• P. Sabinin and M. G. Stone, Transforming n-gons by Folding the Plane, Amer. Math. Monthly, Vol. 102, No. 7 (1995), pp. 620-627.
• Eric Weisstein's World of Mathematics, Folding.
• Robert G. Wilson v, Notes with attachment.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A000217, A001651, A032766, A035361, A063574, A132462.

Complement of A016777.

Range of A002517.

Cf. A274406. [Bruno Berselli, Jun 26 2016]

Programs

Haskell

a007494 =  flip div 2 . (+ 1) . (* 3) -- Reinhard Zumkeller, Dec 12 2014

Magma

[(6*n+1)/4-(-1)^n/4: n in [0..80]]; // Vincenzo Librandi, Aug 20 2011

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 *)

PARI

a(n)=n+(n+1)>>1 \\ Charles R Greathouse IV, Jul 25 2011

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

A145389(a(n)) <> 1. - Reinhard Zumkeller, Oct 10 2008

a(n) = A002943(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010

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

E.g.f.: (3*exp(x)*x + sinh(x))/2. - Stefano Spezia, Feb 11 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - Amiram Eldar, Dec 04 2021