The familiar Fibonacci sequence can be extended easily to negative indices by the relation F(n-1) = F(n+1) - F(n). It is also possible to create a new (double-ended) Fibonacci-like sequence G from any ordered integer pair (a, b) by setting G(0) = a, G(1) = b, G(n) + G(n+1) = G(n+2) for all integers n. It may be the case that for two distinct integer pairs (a, b) and (c, d) that they define the same sequence with shifted indices.

Find an equation involving integers a, b, c and d that is true iff (a, b) and (c, d) define the same sequence(but perhaps shifted).

BONUS(I haven't solved this one):Find a Diophantine equation that fulfills the requirements above.

## Are they in the same sequence?

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### Re: Are they in the same sequence?

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- eta oin shrdlu
**Posts:**450**Joined:**Sat Jan 19, 2008 4:25 am UTC

### Re: Are they in the same sequence?

Partial solution:Edit 5/31 to add another approach:

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