A164360 Period 3: repeat [5, 4, 3].

5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3

Offset: 0

Author

Stephen Crowley, Aug 14 2009

Keywords

• easy
• nonn

Comments

From Klaus Brockhaus, May 29 2010: (Start)

Continued fraction expansion of (32+sqrt(1297))/13.

Decimal expansion of 181/333. (End)

Links

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

Crossrefs

Cf. A007877 (repeat 0,1,2,1), A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2), A158289 (repeat 0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1).

Cf. A178566 (decimal expansion of (32+sqrt(1297))/13). [Klaus Brockhaus, May 29 2010]

Programs

Magma

[ n le 3 select 6-n else Self(n-3):n in [1..105] ]; // Klaus Brockhaus, Sep 17 2009

Magma

&cat [[5, 4, 3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

Maple

seq(op([5, 4, 3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

Mathematica

PadRight[{}, 100, {5, 4, 3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

Formula

a(n) = 4+(-1)^n*((1/2+I*sqrt(3)/6)*((1+I*sqrt(3))/2)^n+(1/2-I*sqrt(3)/6)*((1-I*sqrt(3))/2)^n). [Corrected by Klaus Brockhaus, Sep 17 2009]

a(n) = 4+(1/3)*sqrt(3)*sin(2*n*Pi/3)+cos(2*n*Pi/3). [Corrected by Klaus Brockhaus, Sep 17 2009]

a(n) = a(n-3) for n > 2, with a(0) = 5, a(1) = 4, a(2) = 3.

G.f.: (5+4*x+3*x^2)/((1-x)*(1+x+x^2)). [Klaus Brockhaus, Sep 17 2009]

E.g.f.: 4*exp(x)+(1/3)*sqrt(3)*exp(-(1/2)*x)*sin((1/2)*x*sqrt(3))+exp(-(1/2)*x)*cos((1/2)*x*sqrt(3)).

a(n) = 4 + A057078(n). - Wesley Ivan Hurt, Jul 01 2016

Extensions

Edited by Klaus Brockhaus, Sep 17 2009

Offset changed to 0 and formulas adjusted by Klaus Brockhaus, May 18 2010