A047223 Numbers that are congruent to {1, 2, 3} mod 5.
1, 2, 3, 6, 7, 8, 11, 12, 13, 16, 17, 18, 21, 22, 23, 26, 27, 28, 31, 32, 33, 36, 37, 38, 41, 42, 43, 46, 47, 48, 51, 52, 53, 56, 57, 58, 61, 62, 63, 66, 67, 68, 71, 72, 73, 76, 77, 78, 81, 82, 83, 86, 87, 88, 91, 92, 93, 96, 97, 98, 101, 102, 103, 106, 107
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Crossrefs
Cf. A001622.
Programs
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Magma
[n : n in [0..150] | n mod 5 in [1..3]]; // Wesley Ivan Hurt, Jun 14 2016
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Maple
A047223:=n->(15*n-12-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047223(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016
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Mathematica
Select[Range[100], MemberQ[{1, 2, 3}, Mod[#,5]]&] (* Harvey P. Dale, Oct 28 2013 *) LinearRecurrence[{1, 0, 1, -1}, {1, 2, 3, 6}, 100] (* Vincenzo Librandi, Jun 15 2016 *) -
PARI
a(n)=(n-1)\3*5+n%5 \\ Charles R Greathouse IV, Dec 22 2011
Formula
G.f.: x*(1+x+x^2+2*x^3)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011
a(n) = 2*floor((n-1)/3)+n. - Gary Detlefs, Dec 22 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (15*n-12-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 5k-2, a(3k-1) = 5k-3, a(3k-2) = 5k-4. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2+2/sqrt(5))*Pi/10 - log(phi)/sqrt(5) + 3*log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 16 2023