Search: id:A139011
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%I A139011
%S A139011 2,3,2,7,38,117,278,527,718,237,2642,11753,33802,76443,136762,164833,
%T A139011 24478,922077,3565918,9653287,20783558,34867797,35553398,32125393,
%U A139011 306268562,1064447283,2726446322,5583548873,8701963882
%V A139011 2,3,2,-7,-38,-117,-278,-527,-718,-237,2642,11753,33802,76443,136762,164833,
               -24478,
%W A139011 -922077,-3565918,-9653287,-20783558,-34867797,-35553398,32125393,306268562,
               1064447283,
%X A139011 2726446322,5583548873,8701963882
%N A139011 Real part of (2 + I)^n.
%C A139011 (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.
%C A139011 Imaginary part of (2 + I)^n = A099456
%F A139011 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.
%F A139011 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
%F A139011 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
%F A139011 G.f.: exp(x)^2*cos(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 06 2009]
%F A139011 a(n)=(1/2)*(2-I)^n+1/2*(2+I)^n - Vim Wenders (vim(AT)gmx.li), Apr 12 
               2008
%e A139011 a(5) = -38 since (2 + I)^5 = (-38 + 41*I).
%e A139011 a(5) = -38 since [2,-1; 1,2]^5 = [ -38,-41; 41,-38], where 41 = A139012(5).
%e A139011 a(5) = -38 = A006496(5).
%p A139011 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]
%o A139011 sage: [lucas_number2(n,4,5)/2 for n in xrange(1,31)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 08 2008
%Y A139011 Cf. A099456, A006495, A006496.
%Y A139011 Sequence in context: A014784 A048601 A008317 this_sequence A152297 A063708 
               A096488
%Y A139011 Adjacent sequences: A139008 A139009 A139010 this_sequence A139012 A139013 
               A139014
%K A139011 sign
%O A139011 1,1
%A A139011 Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 05 2008
%E A139011 Cross-reference corrected by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 06 2009

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