OFFSET
0,3
COMMENTS
Equivalently: numerator of real part of (omega)^(-n) where omega = -1 + i*3. - Harvey P. Dale, Sep 14 2013
FORMULA
a(n) = numerator( Re(1/(-1 + i*3)^n) ). 1/(-1 + i*3)^n = A124869(n)/ A124870(n) + i*A124871(n)/A124872(n).
G.f.: (10+x)/(10+2*x+x^2) = 1-1/10*x-2/25*x^2+13/500*x^3+7/2500*x^4-79/25000*x^5+... . - Vladeta Jovovic, Oct 08 2007
Conjectures from Colin Barker, Jul 16 2019: (Start)
G.f.: (1 - x - 2*x^2 + 13*x^3 - 7*x^4 - 65*x^5 + 39*x^6 + 125*x^7 - 625*x^10) / ((1 - 8*x^2 + 25*x^4)*(1 + 8*x^2 + 25*x^4)).
a(n) = 14*a(n-4) - 625*a(n-8) for n>10.
(End)
EXAMPLE
a(0) = 1 = numerator of Re((-1+3*i)^0) = 1/1 + 0*i.
a(1) = -1 = numerator of Re(1/(-1+3*i)) = -1/10 - i*3/10.
a(2) = -2 = numerator of Re((-1+3*i)^(-2)) = -2/25 + i*3/50.
a(3) = 13 = numerator of Re((-1+3*i)^(-3)) = 13/500 + i*9/500.
a(4) = 7 = numerator of Re((-1+3*i)^(-4)) = 7/2500 - i*6/625.
a(5) = -79 = numerator of Re((-1+3*i)^(-5)) = -79/25000 + i*3/25000.
a(6) = 11 = numerator of Re((-1+3*i)^(-6)) = 11/31250 + i*117/125000.
a(7) = 307 = numerator of Re((-1+3*i)^(-7)) = 307/1250000 - i*249/1250000.
a(8) = -527 = numerator of Re((-1+3*i)^(-8)) = -527/6250000 - i*21/390625.
MATHEMATICA
With[{o=-1+3I}, Table[Numerator[Re[o^-n]], {n, 0, 30}]] (* Harvey P. Dale, Sep 14 2013 *)
CROSSREFS
KEYWORD
easy,frac,sign
AUTHOR
Jonathan Vos Post, Nov 11 2006
EXTENSIONS
Removed square roots from definition and formula. - R. J. Mathar, May 02 2009
STATUS
approved