A047206 Numbers that are congruent to {1, 3, 4} mod 5.

1, 3, 4, 6, 8, 9, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108

Offset

1,2

Links

Table of n, a(n) for n=1..65.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

G.f.: (1+2*x+x^2+x^3)/((1-x)^2*(1+x+x^2)).

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = 1+(5*n)/3-(i*sqrt(3) * (-1/2+(i*sqrt(3))/2)^n)/9+(i*sqrt(3)* (-1/2-(i*sqrt(3))/2)^n)/9. - Stephen Crowley, Feb 11 2007

a(n) = floor((5*n-1)/3). - Gary Detlefs, May 14 2011

From Wesley Ivan Hurt, Jun 14 2016: (Start)

a(n) = (15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 5k-1, a(3k-1) = 5k-2, a(3k-2) = 5k-4. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5-sqrt(5))/2)*Pi/5 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 16 2023

Maple

A047206:=n->(15*n-6-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9: seq(A047206(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Mathematica

Select[Range[0, 200], MemberQ[{1, 3, 4}, Mod[#, 5]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *)

Prog

(Magma) [ n : n in [1..150] | n mod 5 in [1, 3, 4] ]; // Vincenzo Librandi, Mar 31 2011
(PARI) a(n)=(5*n-1)\3 \\ Charles R Greathouse IV, Jul 01 2013

Crossrefs

Cf. A001622.

Sequence in context: A059535 A061402 A330066 * A187474 A081031 A285681

Adjacent sequences: A047203 A047204 A047205 * A047207 A047208 A047209

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved