%I #20 Nov 20 2024 15:34:38

%S 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,

%T -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,

%U 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

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

%C 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).

%C 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

%H Holly Krieger and Brady Haran, <a href="https://www.youtube.com/watch?v=N92w4e-hrA4">A Fascinating Thing about Fractions</a>, Numberphile video, Dec 15 2019.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-1).

%F 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]

%F a(n) = cos(n*Pi) + 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3). - _Wesley Ivan Hurt_, Jun 20 2016

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

%t PadRight[{}, 100, {7, 5, 1, -7, -5, -1}] (* _Wesley Ivan Hurt_, Jun 20 2016 *)

%o (Magma) &cat [[7, 5, 1, -7, -5, -1]^^20]; // _Wesley Ivan Hurt_, Jun 20 2016

%Y Cf. A146501, A153130.

%K sign,easy

%O 0,1

%A _Paul Curtz_, Jan 16 2009

%E Edited and extended by _R. J. Mathar_, Jan 23 2009