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

[David Wilson]

The n with sF(n) <= 4 form an obvious pattern, see below.
This pattern includes no sF(n) = 4.
I suspect that you can use an inductive proof to show this pattern is correct and conclude that indeed no sF(n) = 4.

n      sF(n)
0       0
1       1
2       1
3       2
4       1
5       3
6       2
8       1
10      3
12      2
16      1
20      3
24      2
32      1
40      3
48      2
64      1
80      3
96      2
128     1
160     3
192     2

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Alonso
> Del Arte
> Sent: Thursday, March 23, 2017 11:02 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Is 4 a semi-Fibonacci number?
> 
> 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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> <https://www.smashwords.com/profile/view/AlonsoDelarte>
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