[seqfan] Re: Smallest index of Fibonacci-like sequence containing n

[Charles Greathouse]

I think this sequence is interesting. Some quick observations: 0 is the
only number with index 0, all positive integers have index at least 1. If a
number is k times a Fibonacci number, then its index is at most k via (A,
B) = (0, k); in particular, since 1 is a Fibonacci number, a(n) <= n. (This
suggests A054495 as a cross-reference.)

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Wed, Nov 5, 2014 at 1:50 PM, Allan Wechsler <acwacw at gmail.com> wrote:

> Any two non-negative integers can seed a Fibonacci-like sequence, F[0] = A,
> F[1] = B, F[i+2] = F[i+1] + F[i].
>
> Let A+B be called the "index" of this sequence.
>
> Of all Fibonacci-like sequences containing, say, 18, the one with the
> smallest index is {2,1,3,4,7,11,18...}, with an index of 3. So I say A[18]
> = 3.
>
> If n is a classic Fibonacci number, A[n] = 1. If n is a Lucas number (like
> 18), then A[n] = 3. If n is twice a Fibonacci number (like 16) then A[n] =
> 2.
>
> I have calculated A[n] by hand for n from 1 to 24. It is quite possible
> that I have made mistakes, but the sequence I get is:
>
> {1,1,1,2,1,2,3,1,3,2,3,4,1,4,3,2,5,3,5,4,1,6,4,3, ...}
>
> This is not in OEIS. I would've submitted it, but I would like somebody
> else to check my arithmetic first, because it seems unlikely that such a
> simple concept wouldn't have been entered already. If nobody steps up to
> the place quickly I will cobble together some code and submit anyway.
> Thanks for your assistance, seqfans!
>
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