# Representations of Fib by perfect square sequences

creigh at o2online.de creigh at o2online.de
Tue Sep 28 00:16:40 CEST 2004

```Hi. I would like to give two results I find could be of
interest.

CLAIM I:

Define
(a(n)) = A007655 =   (1 ,14,195,2716,37829,526890,7338631,102213944,1423656585  )
// - jes/4
(b(n)) = (1, 16, 225, 3136, 43681, 608400, 8473921, 118026496, 1643897025,
)  // les
(c(n)) = (4, 49, 676, 9409, 131044, 1825201, 25421764, 354079489, 4931691076,
)  // tes
(d(n)) = (1, 9, 121, 1681, 23409, 326041, 4541161, 63250209, 880961761, 12270214441,
) //ves

then for all n:
b(n), c(n), d(n) are perfect squares; sqrt(b(n)) = A001353(n+1)  (Note:
this was "jes" from
http://www.research.att.com/projects/OEIS?Anum=A097947  ! );
sqrt(c(n)) = A001075(n+1)
sqrt(d(n)) = A001835(n+1)

and b(n) + c(n) = d(n) + 4a(n),
[4(a(n))]^2 = b(n+1);

I haven't had time to check all properties given at
http://www.research.att.com/projects/OEIS?Anum=A097947
but I suspect they are very similar. However, before we had for example:
ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1 are perfect squares.
I no
longer see this property showing up (perhaps someone else can find a combination
to produce something like this...?).

CLAIM II:

(a(n)) = (5/2)Fib(12n+12) = (360, 115920, 37325880, 12018817440, )
(e(n)) = (400, 129600, 41731600, 13437446400, 4326816010000, )
(f(n))  = (81, 25921, 8346321, 2687489281, 865363202001, )
(g(n)) = (841, 271441, 87403801, 28143753121, 9062201101801, )

then for all n:
e(n), f(n), g(n) are perfect squares and

(5/2)Fib(12n) + e(n) + f(n) = g(n)

Sincerely,
Creighton

```