A154870 Period 6: repeat [7, 5, 1, -7, -5, -1].

7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1, -7, -5, -1, 7, 5, 1

Offset: 0

Author

Paul Curtz, Jan 16 2009

Keywords

• sign
• easy

Comments

The sequence b(n) = (-A153130(n)) mod 9 = A153130(n+3) = A146501(n-1) = 8, 7, 5, 1, 2, 4,... has period length 6. This here is a(n)=b(n)-A153130(n).

a(n) is (-1)^(n+1) * numerator of F(n) where F(n) = f(F(n-1)) starting from F(0) = -7/4 and step f(z) = z^2 -29/16. - Nicolas Bělohoubek, Nov 20 2024

Links

• Holly Krieger and Brady Haran, A Fascinating Thing about Fractions, Numberphile video, Dec 15 2019.
• Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Crossrefs

Cf. A146501, A153130.

Programs

Magma

&cat [[7, 5, 1, -7, -5, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

Maple

A154870:=n->[7, 5, 1, -7, -5, -1][(n mod 6)+1]: seq(A154870(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

Mathematica

PadRight[{}, 100, {7, 5, 1, -7, -5, -1}] (* Wesley Ivan Hurt, Jun 20 2016 *)

Formula

a(n) = -a(n-3) for n>2; G.f.: (7+5*x+x^2)/((1+x)*(1-x+x^2)). [R. J. Mathar, Jan 23 2009]

a(n) = cos(n*Pi) + 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3). - Wesley Ivan Hurt, Jun 20 2016

Extensions

Edited and extended by R. J. Mathar, Jan 23 2009