0,3

For the generalized Fibonacci sequences U(n-1;a):=(ap(a)^n - am(a)^n)/(ap(a)-am(a)) with ap(a):=(a+sqrt(a^2+4))/2, am(a):=(a-sqrt(a^2+4))/2, a from the integers, one has for the squared sequences U(n-1;a)^2 = (2*T(n,(a^2+2)/2) - 2*(-1)^n)/(a^2+4). Here T(n,x) are Chebyshev's polynomials of the first kind (see A053120). Therefore the o.g.f. for the squared sequence is x*(1-x)/((1-(a^2+2)*x+x^2)*(1+x)) = x*(1-x)/(1-(a^2+1)*x-(a^2+1)*x^2+x^3). For this example a=4.

Unsigned member r=-16 of the family of Chebyshev sequences S_r(n) defined in A092184.

((-1)^(n+1))*a(n) = S_{-16}(n), n>=0, defined in A092184.

Index entries for sequences related to Chebyshev polynomials.

a(n)= A001076(n)^2.

a(n)= 17*a(n-1) + 17*a(n-2) - a(n-3), n>=3; a(0)=0, a(1)=1, a(2)=16.

a(n)= 18*a(n-1) - a(n-2) - 2*(-1)^n, n>=2; a(0)=0, a(1)=1.

a(n)= (T(n, 9)-(-1)^n)/10 with Chebyshev's T(n, x) polynomials of the first kind. T(n, 9)=A023039(n).

G.f.: x*(1-x)/((1-18*x+x^2)*(1+x)) = x*(1-x)/(1-17*x-17*x^2+x^3).

with (combinat):seq(fibonacci(n, 4)^2, n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 09 2008

(Mupad) numlib::fibonacci(3*n)^2/4 $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008

Cf. A007598, A079291, A092936, A099365-6 (other square sequences of this type).

Sequence in context: A004382 A038758 A027776 this_sequence A039746 A053856 A027547

Adjacent sequences: A099276 A099277 A099278 this_sequence A099280 A099281 A099282

nonn,easy

Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004