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A047240
Numbers that are congruent to {0, 1, 2} mod 6.
13
0, 1, 2, 6, 7, 8, 12, 13, 14, 18, 19, 20, 24, 25, 26, 30, 31, 32, 36, 37, 38, 42, 43, 44, 48, 49, 50, 54, 55, 56, 60, 61, 62, 66, 67, 68, 72, 73, 74, 78, 79, 80, 84, 85, 86, 90, 91, 92, 96, 97, 98, 102, 103, 104, 108, 109, 110, 114, 115, 116, 120, 121, 122
OFFSET
1,3
COMMENTS
Partial sums of 0,1,1,4,1,1,4,... - Paul Barry, Feb 19 2007
Numbers k such that floor(k/3) = 2*floor(k/6). - Bruno Berselli, Oct 05 2017
FORMULA
From Paul Barry, Feb 19 2007: (Start)
G.f.: x*(1 + x + 4*x^2)/((1 - x)*(1 - x^3)).
a(n) = 2*n - 3 - cos(2*n*Pi/3) + sin(2*n*Pi/3)/sqrt(3). (End)
a(n) = n-1 + 3*floor((n-1)/3). - Philippe Deléham, Apr 21 2009
a(n) = 6*floor(n/3) + (n mod 3). - Gary Detlefs, Mar 09 2010
a(n+1) = Sum_{k>=0} A030341(n,k)*b(k) with b(0)=1 and b(k)=2*3^k for k>0. - Philippe Deléham, Oct 22 2011.
a(n) = 2*n - 2 - A010872(n-1). - Wesley Ivan Hurt, Jul 07 2013
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(3*k) = 6*k-4, a(3*k-1) = 6*k-5, a(3*k-2) = 6*k-6. (End)
Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(3))*Pi/18 + log(2+sqrt(3))/(2*sqrt(3)). - Amiram Eldar, Dec 14 2021
E.g.f.: 4 + exp(x)*(2*x - 3) - exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Jul 26 2024
MAPLE
A047240:=n->2*n-3-cos(2*n*Pi/3)+sin(2*n*Pi/3)/sqrt(3): seq(A047240(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016
select(k -> modp(iquo(k, 3), 2) = 0, [$0..122]); # Peter Luschny, Oct 05 2017
MATHEMATICA
Select[Range[0, 200], Mod[#, 6] == 0 || Mod[#, 6] == 1 || Mod[#, 6] == 2 &] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
a047240[n_] := Flatten[Map[6 # + {0, 1, 2} &, Range[0, n]]]; a047240[20] (* data *) (* Hartmut F. W. Hoft, Mar 06 2017 *)
PROG
(Magma) [0], [6*Floor(n/3) + (n mod 3): n in [1..65]]; // Vincenzo Librandi, Oct 23 2011
(PARI) a(n)=n\3*6 + n%3 \\ Charles R Greathouse IV, Oct 07 2015
(Python 3)
[k for k in range(123) if (k//3) % 2 == 0] # Peter Luschny, Oct 05 2017
CROSSREFS
Cf. similar sequences with formula n+i*floor(n/3) listed in A281899.
Sequence in context: A201819 A275523 A327258 * A080333 A194369 A039592
KEYWORD
nonn,easy
EXTENSIONS
Paul Barry formula adapted for offset 1 by Wesley Ivan Hurt, Jun 14 2016
STATUS
approved