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

Allan Wechsler acwacw at gmail.com
Wed Nov 5 19:50:28 CET 2014

Any two non-negative integers can seed a Fibonacci-like sequence, F = A,
F = 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
= 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.