1,1

(a(n))^2 + (A139012(n))^2 = 5^n = 3125. Example: (a(5))^2 + (A139012(5))^2 = 3125 = 5^5 = (-38)^2 + 41^2 = 1444 + 1681.

Imaginary part of (2 + I)^n = A099456

Real part of (2 + I)^n, I^2 = -1. Term (1,1) of matrix [2,-1; 1,2]^n. Irrespective of signs, odd indexed terms of A006496 interleaved with even indexed signs of A006495.

O.g.f.: x(2-5x)/(1-4x+5x^2). a(n)=4*a(n-1)-5*a(n-2) = 2*A099456(n-1)-5*A099456(n-2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 06 2008

a(n)=[1-(1/2)*I]*(2-I)^n+[1+(1/2)*I]*(2+I)^n, with n>=0 and I=sqrt(-1) - Paolo P. Lava (ppl(AT)spl.at), Jul 14 2008

G.f.: exp(x)^2*cos(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 06 2009]

a(n)=(1/2)*(2-I)^n+1/2*(2+I)^n - Vim Wenders (vim(AT)gmx.li), Apr 12 2008

a(5) = -38 since (2 + I)^5 = (-38 + 41*I).

a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A139012(5).

a(5) = -38 = A006496(5).

restart: G(x):=exp(x)^2*cos(x): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=1..29 ); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 06 2009]

sage: [lucas_number2(n, 4, 5)/2 for n in xrange(1, 31)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2008

Cf. A099456, A006495, A006496.

Adjacent sequences: A139008 A139009 A139010 this_sequence A139012 A139013 A139014

Sequence in context: A014784 A048601 A008317 this_sequence A152297 A063708 A096488

sign

Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 05 2008

Cross-reference corrected by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jan 06 2009