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%I #77 Sep 13 2023 09:37:27
%S 0,1,2,1,4,5,2,7,8,3,10,11,4,13,14,5,16,17,6,19,20,7,22,23,8,25,26,9,
%T 28,29,10,31,32,11,34,35,12,37,38,13,40,41,14,43,44,15,46,47,16,49,50,
%U 17,52,53,18,55,56,19,58,59,20,61,62,21,64,65,22,67
%N If n mod 3 = 0 then n/3 else n.
%C Numerator of n/3. - _Wesley Ivan Hurt_, Jul 18 2014
%H Reinhard Zumkeller, <a href="/A051176/b051176.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F a(n) = n / gcd(n,3).
%F G.f.: x*(1+2*x+x^2+2*x^3+x^4)/(1-x^3)^2 = x*(1+2*x+x^2+2*x^3+x^4) / ( (x-1)^2*(1+x+x^2)^2 ). - _Len Smiley_, Apr 30 2001
%F Multiplicative with a(3^e) = 3^(e-1), a(p^e) = p^e otherwise. - _Mitch Harris_, Jun 09 2005
%F a(n) = A167192(n+3, 3). - _Reinhard Zumkeller_, Oct 30 2009
%F From _R. J. Mathar_, Apr 18 2011: (Start)
%F a(n) = A109044(n)/3.
%F Dirichlet g.f.: zeta(s-1)*(1-2/3^s). (End)
%F a(n) = n/3 * (1 + 2*A011655(n)) = n*A144437(n)/3. - _Timothy Hopper_, Feb 23 2017
%F G.f.: x /(1 - x)^2 - 2 * x^3/(1 - x^3)^2. - _Michael Somos_, Mar 05 2017
%F a(n) = a(-n) for all n in Z. - _Michael Somos_, Mar 05 2017
%F a(n) = n*(7 - 4*cos((2*Pi*n)/3)) / 9. - _Colin Barker_, Mar 05 2017
%F Sum_{k=1..n} a(k) ~ (7/18) * n^2. - _Amiram Eldar_, Nov 25 2022
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(2)/3. - _Amiram Eldar_, Sep 08 2023
%e G.f. = x + 2*x^2 + x^3 + 4*x^4 + 5*x^5 + 2*x^6 + 7*x^7 + 8*x^8 + 3*x^9 + ...
%p A051176:=n->numer(n/3); seq(A051176(n), n=0..100); # _Wesley Ivan Hurt_, Jul 18 2014
%t If[Divisible[#,3],#/3,#]&/@Range[0,70] (* _Harvey P. Dale_, Feb 07 2011 *)
%t a[n_] := Numerator[n/3]; Array[a, 100, 0] (* _Wesley Ivan Hurt_, Jul 18 2014 *)
%o (Haskell)
%o a051176 n = if m == 0 then n' else n where (n',m) = divMod n 3
%o -- _Reinhard Zumkeller_, Aug 27 2012
%o (PARI) a(n) = if (n % 3, n, n/3); \\ _Michel Marcus_, Feb 02 2016
%o (Magma) [Numerator(n/3): n in [0..70]]; // _G. C. Greubel_, Feb 19 2019
%o (Sage) [numerator(n/3) for n in range(70)] # _G. C. Greubel_, Feb 19 2019
%Y Cf. A026741, A051176, A060819, A060791, A060789 for n / GCD(n,k) for k=2..6. See also A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).
%Y Cf. A109044, A011655, A144437, A167192.
%K nonn,easy,mult
%O 0,3
%A _N. J. A. Sloane_
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