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A124869 Numerator of real part of (2*omega)^(-n) where omega = (-1 + i*3)/ 2. 4

%I

%S 1,-1,-2,13,7,-79,11,307,-527,-481,779,-3827,-11753,42641,4031,

%T -245453,164833,897599,-430441,-1044467,9653287,-14084239,-8545549,

%U 138785587,32125393,-758178721,149387939,2595790093,-5583548873,-1811852719

%N Numerator of real part of (2*omega)^(-n) where omega = (-1 + i*3)/ 2.

%C Equivalently: numerator of real part of (omega)^(-n) where omega = -1 + i*3. - _Harvey P. Dale_, Sep 14 2013

%F a(n) = numerator( Re(1/(-1 + i*3)^n) ). 1/(-1 + i*3)^n = A124869(n)/ A124870(n) + i*A124871(n)/A124872(n).

%F 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

%F Conjectures from _Colin Barker_, Jul 16 2019: (Start)

%F 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)).

%F a(n) = 14*a(n-4) - 625*a(n-8) for n>10.

%F (End)

%e a(0) = 1 = numerator of Re((-1+3*i)^0) = 1/1 + 0*i.

%e a(1) = -1 = numerator of Re(1/(-1+3*i)) = -1/10 - i*3/10.

%e a(2) = -2 = numerator of Re((-1+3*i)^(-2)) = -2/25 + i*3/50.

%e a(3) = 13 = numerator of Re((-1+3*i)^(-3)) = 13/500 + i*9/500.

%e a(4) = 7 = numerator of Re((-1+3*i)^(-4)) = 7/2500 - i*6/625.

%e a(5) = -79 = numerator of Re((-1+3*i)^(-5)) = -79/25000 + i*3/25000.

%e a(6) = 11 = numerator of Re((-1+3*i)^(-6)) = 11/31250 + i*117/125000.

%e a(7) = 307 = numerator of Re((-1+3*i)^(-7)) = 307/1250000 - i*249/1250000.

%e a(8) = -527 = numerator of Re((-1+3*i)^(-8)) = -527/6250000 - i*21/390625.

%t With[{o=-1+3I},Table[Numerator[Re[o^-n]],{n,0,30}]] (* _Harvey P. Dale_, Sep 14 2013 *)

%K easy,frac,sign

%O 0,3

%A _Jonathan Vos Post_, Nov 11 2006

%E Removed square roots from definition and formula. - _R. J. Mathar_, May 02 2009

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Last modified April 19 15:36 EDT 2021. Contains 343116 sequences. (Running on oeis4.)