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

%I A007494 #157 Feb 16 2025 08:32:31
%S A007494 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,
%T A007494 38,39,41,42,44,45,47,48,50,51,53,54,56,57,59,60,62,63,65,66,68,69,71,
%U A007494 72,74,75,77,78,80,81,83,84,86,87,89,90,92,93,95,96,98,99,101,102,104,105,107
%N A007494 Numbers that are congruent to 0 or 2 mod 3.
%C A007494 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
%C A007494 Partial sums of 0,2,1,2,1,2,1,2,1,... . - _Paul Barry_, Aug 18 2007
%C A007494 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
%C A007494 Range of A173732. - _Reinhard Zumkeller_, Apr 29 2012
%C A007494 Number of partitions of 6n into two odd parts. - _Wesley Ivan Hurt_, Nov 15 2014
%C A007494 Numbers m such that 3 divides A000217(m). - _Bruno Berselli_, Aug 04 2017
%C A007494 Maximal length of a snake like polyomino that fits in a 2 X n rectangle. - _Alain Goupil_, Feb 12 2020
%D A007494 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.
%H A007494 Vincenzo Librandi, <a href="/A007494/b007494.txt">Table of n, a(n) for n = 0..10000</a>
%H A007494 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1002">Encyclopedia of Combinatorial Structures 1002</a>.
%H A007494 Attila Máder, <a href="http://www.model.u-szeged.hu/etc/edoc/imp/AMader/AMader.pdf">The Use of Experimental Mathematics in the Classroom</a>, in Interesting Mathematical Problems in Sciences and Everyday Life - 2011.
%H A007494 Kurt Mahler, <a href="https://carmamaths.org/resources/mahler/docs/167.pdf">An unsolved problem on the powers of 3/2</a>, J. Austral. Math. Soc., Vol. 8 (1968), pp. 313-321.
%H A007494 P. Sabinin and M. G. Stone, <a href="http://www.jstor.org/stable/2974558">Transforming n-gons by Folding the Plane</a>, Amer. Math. Monthly, Vol. 102, No. 7 (1995), pp. 620-627.
%H A007494 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Folding.html">Folding</a>.
%H A007494 Robert G. Wilson v, <a href="/A007494/a007494.pdf">Notes with attachment</a>.
%H A007494 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A007494 a(n) = 3*n/2 if n even, otherwise (3*n+1)/2.
%F A007494 If u(1)=0, u(n) = n + floor(u(n-1)/3), then a(n-1) = u(n). - _Benoit Cloitre_, Nov 26 2002
%F A007494 G.f.: x*(x+2)/((1-x)^2*(1+x)). - _Ralf Stephan_, Apr 13 2002
%F A007494 a(n) = 3*floor(n/2) + 2*(n mod 2) = A032766(n) + A000035(n). - _Reinhard Zumkeller_, Apr 04 2005
%F A007494 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
%F A007494 A145389(a(n)) <> 1. - _Reinhard Zumkeller_, Oct 10 2008
%F A007494 a(n) = A002943(n) - A173511(n). - _Reinhard Zumkeller_, Feb 20 2010
%F A007494 a(n) = 3*n - a(n-1) - 1 (with a(0)=0). - _Vincenzo Librandi_, Nov 18 2010
%F A007494 a(n) = Sum_{k>=0} A030308(n,k)*A042950(k). - _Philippe Deléham_, Oct 17 2011
%F A007494 a(n) = n + ceiling(n/2). - _Arkadiusz Wesolowski_, Sep 18 2012
%F A007494 a(n) = 2n - floor(n/2) = floor((3n+1)/2) = n + (n + (n mod 2))/2. - _Wesley Ivan Hurt_, Oct 19 2013
%F A007494 a(n) = A000217(n+1) - A099392(n+1). - _Bui Quang Tuan_, Mar 27 2015
%F A007494 a(n) = n + floor(n/2) + (n mod 2). - _Bruno Berselli_, Apr 04 2016
%F A007494 a(n) = Sum_{i=1..n} numerator(2/i). - _Wesley Ivan Hurt_, Feb 26 2017
%F A007494 a(n) = Sum_{k=0..n-1} Sum_{i=0..k} C(i,k)+(-1)^(k-i). - _Wesley Ivan Hurt_, Sep 20 2017
%F A007494 E.g.f.: (3*exp(x)*x + sinh(x))/2. - _Stefano Spezia_, Feb 11 2020
%F A007494 Sum_{n>=1} (-1)^(n+1)/a(n) = log(3)/2 - Pi/(6*sqrt(3)). - _Amiram Eldar_, Dec 04 2021
%p A007494 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
%p A007494 A007494:=n->floor((3*n+1)/2); seq(A007494(k), k=0..100); # _Wesley Ivan Hurt_, Sep 27 2013
%t A007494 Flatten[{#,#+2}&/@(3Range[0,40])] (* _Harvey P. Dale_, May 15 2011 *)
%t A007494 Table[2n - Floor[n/2], {n,0,100}] (* _Wesley Ivan Hurt_, Sep 27 2013 *)
%o A007494 (PARI) a(n)=n+(n+1)>>1 \\ _Charles R Greathouse IV_, Jul 25 2011
%o A007494 (Magma) [(6*n+1)/4-(-1)^n/4: n in [0..80]]; // _Vincenzo Librandi_, Aug 20 2011
%o A007494 (Haskell)
%o A007494 a007494 =  flip div 2 . (+ 1) . (* 3) -- _Reinhard Zumkeller_, Dec 12 2014
%Y A007494 Cf. A000217, A001651, A032766, A035361, A063574, A132462.
%Y A007494 Complement of A016777.
%Y A007494 Range of A002517.
%Y A007494 Cf. A274406. [_Bruno Berselli_, Jun 26 2016]
%K A007494 nonn,easy
%O A007494 0,2
%A A007494 Christopher Lam Cham Kee (Topher(AT)CyberDude.Com)