A090131 Expansion of (1+x)/(1 - 2*x + 2*x^2).

1, 3, 4, 2, -4, -12, -16, -8, 16, 48, 64, 32, -64, -192, -256, -128, 256, 768, 1024, 512, -1024, -3072, -4096, -2048, 4096, 12288, 16384, 8192, -16384, -49152, -65536, -32768, 65536, 196608, 262144, 131072, -262144, -786432, -1048576, -524288, 1048576, 3145728, 4194304, 2097152, -4194304

Offset: 0

Author

Paul Barry, Nov 21 2003

Keywords

• easy
• sign

Comments

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

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-2)

Crossrefs

Cf. A078069.

Programs

Mathematica

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 *)

Formula

a(n) = sum_{k=0..n} C(n, k)(-1)^floor(k/2)(1 + (1 - (-1)^k)/2).

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

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

a(n) = 4a(n-1) - 6a(n-2) + 4a(n-3). - Paul Curtz, Nov 20 2007

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