0,2

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 by sines. Indeed in the complex plan 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

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 (bpcrtz(AT)free.fr), Nov 20 2007

Cf. A078069.

Sequence in context: A084521 A021296 A078069 this_sequence A139525 A133570 A117041

Adjacent sequences: A090128 A090129 A090130 this_sequence A090132 A090133 A090134

easy,sign

Paul Barry (pbarry(AT)wit.ie), Nov 21 2003