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

[Frank Adams-Watters]

This sequence is probably not present, but a number of related 
sequences are. A035506 is a place to start looking.

Franklin T. Adams-Watters

-----Original Message-----
From: Allan Wechsler <acwacw at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Wed, Nov 5, 2014 1:33 pm
Subject: [seqfan] Smallest index of Fibonacci-like sequence containing n


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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