Hopefully, nice(?) sequence relating to Fib, Luc.

Paul D Hanna pauldhanna at juno.com
Tue Sep 21 15:47:12 CEST 2004

For your sequence, I get:

a(n) = ( 2*fibonacci(2*n+3) + fibonacci(2*n-2) )^2

and since 
Lucas(n) = fibonacci(n+2) - fibonacci(n-2) 
Lucas(2*n+2) = fibonacci(2*n+4) - fibonacci(2*n) 

Then I believe that your result could be proved 
using the form:
( 9*fibonacci(2n+2) + Lucas(2n+2))^2 

although I do not have time right now to work it out.

Interesting Fibonacci relation, nonetheless!

-- creigh at o2online.de wrote:

As neither 
(a(n)) = (9, 100, 729, 5041, 34596, 237169, 1625625, 11142244, 76370121, 523448641, )
nor (sqrt(a(n)) 
appear to be listed at OEIS, I'm hoping the following relation will not 
be a trivial consequence of some other formula:

81[Fib(2n+2)]^2 + [Luc(2n+2)]^2 = 4a(n) + 18[Fib(4n+4)]

[by the way, this was a found by slightly modifying the formula for "les, 
ves," etc. at http://www.research.att.com/projects/OEIS?Anum=A097947 ]

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