In One Sequence Or Another
Leroy Quet
qq-quet at mindspring.com
Fri Mar 26 02:45:45 CET 2004
A couple of things:
First, 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.
>...
Hmmm... Do you know what the non-Fibonacci numbers' recursive definition
is?
Does their direct formula involve powers of (3 + sqrt(5))/2 somehow?
(Just a guess. Perhaps the formula does not.)
Second, Marc LeBrun's email on this topic got me to wondering about
sequences derived from other sequences of distinct positive integers via
floor(f(k)) or ceiling(f(k)).
For example, if we have a sequence {a(k)} where this sequence's ordinary
generating function or exponential generating function, f(x),
is monotonically increasing, is positive, and is defined for all positive
x (even if only by analytical continuation for some x's),
then we can have the sequence transforms
b(k) = floor(f(k)),
c(k) = ceiling(f(k)),
d(k) = floor(f(k)) + k,
and
e(k) = ceiling(f(k)) +k-1.
(I include the last 2 because of my result in this thread's original
message.)
thanks,
Leroy Quet
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