A051176 If n mod 3 = 0 then n/3 else n.
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, 28, 29, 10, 31, 32, 11, 34, 35, 12, 37, 38, 13, 40, 41, 14, 43, 44, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 19, 58, 59, 20, 61, 62, 21, 64, 65, 22, 67
Offset: 0
Examples
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 + ...
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Crossrefs
Programs
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Haskell
a051176 n = if m == 0 then n' else n where (n',m) = divMod n 3 -- Reinhard Zumkeller, Aug 27 2012
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Magma
[Numerator(n/3): n in [0..70]]; // G. C. Greubel, Feb 19 2019
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Maple
A051176:=n->numer(n/3); seq(A051176(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2014
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Mathematica
If[Divisible[#,3],#/3,#]&/@Range[0,70] (* Harvey P. Dale, Feb 07 2011 *) a[n_] := Numerator[n/3]; Array[a, 100, 0] (* Wesley Ivan Hurt, Jul 18 2014 *) LinearRecurrence[{0,0,2,0,0,-1},{0,1,2,1,4,5},70] (* Harvey P. Dale, Jul 12 2025 *)
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PARI
a(n) = if (n % 3, n, n/3); \\ Michel Marcus, Feb 02 2016
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Sage
[numerator(n/3) for n in range(70)] # G. C. Greubel, Feb 19 2019
Formula
a(n) = n / gcd(n,3).
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
Multiplicative with a(3^e) = 3^(e-1), a(p^e) = p^e otherwise. - Mitch Harris, Jun 09 2005
a(n) = A167192(n+3, 3). - Reinhard Zumkeller, Oct 30 2009
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109044(n)/3.
Dirichlet g.f.: zeta(s-1)*(1-2/3^s). (End)
G.f.: x /(1 - x)^2 - 2 * x^3/(1 - x^3)^2. - Michael Somos, Mar 05 2017
a(n) = a(-n) for all n in Z. - Michael Somos, Mar 05 2017
a(n) = n*(7 - 4*cos((2*Pi*n)/3)) / 9. - Colin Barker, Mar 05 2017
Sum_{k=1..n} a(k) ~ (7/18) * n^2. - Amiram Eldar, Nov 25 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(2)/3. - Amiram Eldar, Sep 08 2023
Comments