# In One Sequence Or Another

Leroy Quet qq-quet at mindspring.com
Sat Mar 27 18:20:49 CET 2004

```Henry Gould wrote:

>Marc and Leroy - Hey! These complementary sequences are really a lot of fun!
>I wrote a paper styled "Non-Fibonacci numbers", Fibonacci Quart., 3(1965),
>177-183,
>about the sequence complementary to 1, 2, 3, 5, 8, 13, 21, 34, . . .  and
>showed
>how a formula can be gotten. These non-Fibonacci numbers are in the OEIS of
>course.
>...

I should mention the following in this thread:

First, the m_th Fibonacci number (for m = any positive integer) is

floor(phi^m /sqrt(5) +1/2)

(where phi = (sqrt(5)+1)/2, of course).

Let g be a real-> real function such that

x = g(phi^x /sqrt(5) +1/2 -x)

for all x >=0.

Now, g is only single-valued monotonically-increasing for x>0, and
floor(g(0)) = 4.
So, calling 4 the 0-th non-Fibonacci number, we have:

ceiling(g(m)) + m -1

= floor(g(m)) + m       (because g(m) is not an integer)

= the m_th non-Fibonacci,

using my theorem, with some adjustments.

:)

thanks,
Leroy Quet

```