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
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