A131711 Period 12: repeat 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1.

0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1, 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1, 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1, 0, 1, 2, 5, 2, 9, 0, 9, 8, 5, 8, 1

Offset: 0

Author

Paul Curtz, Sep 14 2007

Keywords

• nonn
• easy
• less

Comments

Final digits of Pell numbers. First differences: 1, 1, 3, -3, 7, -9, 9, -1, -3, 3 -7, -1, 1 (cf. A131707).

Can be though of as 2 interlocking sequences, each of the form a(n) = a(n - 1) - a(n - 2) + a(n - 3) - a(n - 4) + a(n - 5).

Links

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

Programs

Mathematica

PadRight[{},120,{0,1,2,5,2,9,0,9,8,5,8,1}] (* Harvey P. Dale, Dec 29 2014 *)

PARI

[0,1,2,5,2,9,0,9,8,5,8,1][n%12+1] \\ Charles R Greathouse IV, Jun 02 2011

Formula

G.f.: (x^8+8x^7+4x^6+5x^4+4x^2+2x+1)x/((1-x) (1+x) (x^2+x+1) (x^2-x+1) (x^4-x^2+1)). a(n) = |A131201(n)| = A000129(n) mod 10 = A000129(n)-10*A131727(n). [From R. J. Mathar, Sep 20 2008]

a(n) = 25/6 -4*cos(Pi*n/6)/sqrt(3) -sqrt(3)*sin(Pi*n/6) -5*cos(Pi*n/3)/3 -5*cos(2*Pi*n/3)/3 +4*cos(5*Pi*n/6)/sqrt(3) +sqrt(3)*sin(5*Pi*n/6) -5*(-1)^n/6. - R. J. Mathar, Oct 08 2011