[seqfan] Re: Is 4 a semi-Fibonacci number?

israel at math.ubc.ca israel at math.ubc.ca
Thu Mar 23 16:23:49 CET 2017

The least n for which any value occurs must be odd, since sF(n) = sF(n/2) 
for even n. So if you have ruled out sF(n) = 4 for odd n, it follows that 4 
can never occur.


On Mar 23 2017, Alonso Del Arte wrote:t

>As you know, 4 is not a Fibonacci number. The Fibonacci function can be
>extended to all real numbers, but to get Fibonacci(x) = 4 requires x be
>what looks like a transcendental number.
>But could 4 be a semi-Fibonacci number? (A030067) The definition is sF(1) =
>1, sF(n) = sF(n/2) if n is even, sF(n) = sF(n - 1) + sF(n - 2) if n is odd.
>Couple of years ago, Roberg G Wilson v determined that 4 does not occur
>among the first million terms. It's not a rigorous proof, of course, but it
>does suggest that 4 never occurs.
>It's fairly easy to prove that sF(n) = 4 is impossible if n is odd. But I
>haven't been able to rule out sF(n) = 4 for n even. That would mean sF(n) =
>8, but the parity of n seems like it could be anything. I can visualize a
>whole tree but many of the branches of that tree might not even exist.
>Can anyone make a determination on this question, or is this another one of
>those plausible but unproven conjectures?

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