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%I #19 Jun 13 2015 00:51:16
%S 1,3,4,2,-4,-12,-16,-8,16,48,64,32,-64,-192,-256,-128,256,768,1024,
%T 512,-1024,-3072,-4096,-2048,4096,12288,16384,8192,-16384,-49152,
%U -65536,-32768,65536,196608,262144,131072,-262144,-786432,-1048576,-524288,1048576,3145728,4194304,2097152,-4194304
%N Expansion of (1+x)/(1 - 2*x + 2*x^2).
%C Also first of two associated sequences a(n) and b(n) built from a(0)=1 and b(0)=2 by the formulas a(n) = a(n-1) + b(n-1) and b(n) = -a(n-1) + b(n-1). The initial terms of the second sequence b(n) are 2, 1, -2, -6, -8, -4, 8, 24, 32, 16, -32, -96, -128, -64, 128, 384, 512, 256, -1536, -2048, -1024, 2048, 6144, 8192, .... The formula for b(n) is the same as for a(n) but replacing cosines with sines. Indeed in the complex plane the points Mn=a(n)+b(n)*I are located where the logarithmic spiral Rho=A*(B^Theta) cuts the two pairs of orthogonal straight lines drawn from the origin with slopes 2, 1/3, -1/2 and -3. - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 29 2007
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2)
%F a(n) = sum_{k=0..n} C(n, k)(-1)^floor(k/2)(1 + (1 - (-1)^k)/2).
%F a(n) = A*(B^Theta(n))*cos(Theta(n)) where A = 3.644691771.. = (5^0,5)*16^(arctan(2)/(2*Pi)) B = 0.64321824.. = 16^(-1/(2*Pi)) Theta(4p+1) = p*Pi + arctan(2) Theta(4*p+2) = p*Pi + arctan(1/3) Theta(4*p+3) = p*Pi + arctan(-1/2) Theta(4*p+4) = p*Pi + arctan(-3). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 29 2007
%F Also a(0)=1, a(1)=3, a(2)=4, a(3)=2 and for n>3 a(n) = -4 * a(n-4). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 29 2007
%F a(n) = 4a(n-1) - 6a(n-2) + 4a(n-3). - _Paul Curtz_, Nov 20 2007
%F a(n) = 2a(n-1) - 2a(n-2) = A009545(n) + A009545(n+1) = (1/2)*((1+2*i)*(1-i)^n + (1-2*i)*(1+i)^n). - _Ralf Stephan_, Jul 21 2013
%t a[n_]:=(MatrixPower[{{1,-1},{1,1}},n].{{2},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%Y Cf. A078069.
%K easy,sign
%O 0,2
%A _Paul Barry_, Nov 21 2003
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