[seqfan] Re: Is 4 a semi-Fibonacci number?
David Wilson
davidwwilson at comcast.net
Fri Mar 24 02:36:36 CET 2017
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
> Author at SmashWords.com
> <https://www.smashwords.com/profile/view/AlonsoDelarte>
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