A158068 Period 6: repeat [1, 2, 2, 1, 5, 5].

1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2

Offset: 0

Author

Paul Curtz, Mar 12 2009

Keywords

• nonn
• easy

Comments

The sequence can be generated starting an array T(n,k) by placing the

periodic sequence 1,2,5 (repeat 1,2,5) in the top row n=0, then defining

the next rows by T(n+1,k) = T(n,k)*T(n,k+1) mod 9, which all have a period T(n,k)=T(n,k+3).

One finds the periodicity T(n+6,k)=T(n,k), and then defines a(n)=T(n,1).

Also the partial fraction expansion of (85+sqrt(12469))/138.

Also the decimal expansion of 11105/90909.

Links

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

Crossrefs

Cf. A157742, A158012.

Programs

Magma

&cat[[1, 2, 2, 1, 5, 5]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

Maple

A158068:=n->(8-5*cos(2*n*Pi/3)-3*sqrt(3)*sin(n*Pi/3))/3: seq(A158068(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016

Mathematica

PadRight[{}, 120, {1,2,2,1,5,5}] (* or *) LinearRecurrence[{1,-1,1,-1,1},{1,2,2,1,5}, 120] (* Harvey P. Dale, Jan 27 2015 *)

Formula

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.

G.f.: (1+x+5*x^4+x^2)/((1-x)*(1-x+x^2)*(1+x+x^2)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009]

a(n) = (8-5*cos(2*n*Pi/3)-3*sqrt(3)*sin(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 17 2016

Extensions

Offset set to 0 - R. J. Mathar, Sep 17 2009