0,3

The n-th rectangle is F(n)*F(n+1), where F(n) = n-th Fibonacci number (F(1)=1, F(2)=1, F(3)=2, etc.), A000045.

If 2*T(a_n) = the oblong number formed by substituting a(n) in the product formula x(x+1), then 2*T(a_n) = F(n-1)*F(n) * F(n)*F(n+1). Thus a(n) equals the integer part of the square root of the right hand side of the given equation. - Kenneth J. Ramsey (RamseyKK2(AT)aol.com), Dec 19 2006

Index entries for sequences related to Chebyshev polynomials.

a(n)=F(n+1)^2 - 1 if n is even, or F(n+1)^2 if n is odd.

G.f.: x/((1-x^2)*(1-3*x+x^2)). - njas Jul 15 2002

a(n)=sum{k=0..floor(n/2), U(n-2k-1, 3/2)} - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

Let M_n denote the n X n Hankel matrix M_n(i, j)=F(i+j-1) where F(n)=A000045(n) is Fibonacci numbers, then the characteristic polynomial of M_n is x^n - F(2n)x^(n-1) + a(n-1)x^(n-2) . - Michael Somos, Nov 14 2002

Table[ Sum[ Fibonacci[k]*Fibonacci[k + 1], {k, 1, n} ], {n, 0, 30} ]

(PARI) a(n)=if(n<0, 0, fibonacci(n+1)^2-1+n%2)

Cf. A000045.

Partial sums of A001654. a(n) = A005313(n+1) - n.

Adjacent sequences: A064828 A064829 A064830 this_sequence A064832 A064833 A064834

Sequence in context: A123888 A097134 A123892 this_sequence A096168 A051042 A121907

nonn

Howard Stern (hsstern(AT)mindspring.com), Oct 23 2001

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 25 2001