A047242 Numbers that are congruent to {0, 1, 3} mod 6.
0, 1, 3, 6, 7, 9, 12, 13, 15, 18, 19, 21, 24, 25, 27, 30, 31, 33, 36, 37, 39, 42, 43, 45, 48, 49, 51, 54, 55, 57, 60, 61, 63, 66, 67, 69, 72, 73, 75, 78, 79, 81, 84, 85, 87, 90, 91, 93, 96, 97, 99, 102, 103, 105, 108, 109, 111, 114, 115, 117, 120, 121, 123
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Programs
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Haskell
a047242 n = a047242_list !! n a047242_list = elemIndices 0 a214090_list -- Reinhard Zumkeller, Jul 06 2012
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Magma
[n-1+Floor((n-1)/3)+Floor((2*n-2)/3) : n in [1..50]]; // Wesley Ivan Hurt, Dec 03 2014
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Maple
A047242:=n->n-1+floor((n-1)/3)+floor((2*n-2)/3): seq(A047242(n), n=1..50); # Wesley Ivan Hurt, Dec 03 2014
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Mathematica
Select[Range[0, 200], Mod[#, 6] == 0 || Mod[#, 6] == 1 || Mod[#, 6] == 3 &] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
Formula
Equals partial sums of (0, 1, 2, 3, 1, 2, 3, 1, 2, 3, ...). - Gary W. Adamson, Jun 19 2008
G.f.: x^2*(1+2*x+3*x^2)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
A214090(a(n)) = 0. - Reinhard Zumkeller, Jul 06 2012
a(n) = a(n-1) + a(n-3) - a(n-4), n>4. - Wesley Ivan Hurt, Dec 03 2014
a(n) = n-1 + floor((n-1)/3) + floor((2n-2)/3). - Wesley Ivan Hurt, Dec 03 2014
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = (6*n-8-cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3.
a(3k) = 6k-3, a(3k-1) = 6k-5, a(3k-2) = 6k-6. (End)
a(n) = 2*n - 2 - sign((n-1) mod 3). - Wesley Ivan Hurt, Sep 26 2017
Sum_{n>=2} (-1)^n/a(n) = Pi/12 + log(2)/6 + log(2+sqrt(3))/(2*sqrt(3)). - Amiram Eldar, Dec 14 2021
E.g.f.: (9 + exp(x)*(6*x - 8) - exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Jul 26 2024