[seqfan] Is 4 a semi-Fibonacci number?

[Alonso Del Arte]

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?

Al

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Alonso del Arte
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