[seqfan] Re: Periodic Fibonacci-like sequences without multiples of several primes

[Vladimir Shevelev]

Thank you very much for this simple remark.  A period of {F_(a,b,...)(n)} could be not multiple of 3 if among a,b,... there is at least one even number. For example, if a=7, b=4 we obtain period of length 8: 0,1,1,2,3,5,2,1,3,1,1,...
  On the other hand, it seems that {F_4(n)} is not periodic (see A224372).

Best regards,
Vladimir


----- Original Message -----
From: Don Reble <djr at nk.ca>
Date: Wednesday, April 3, 2013 18:01
Subject: [seqfan] Re: Periodic Fibonacci-like sequences without multiples of several primes
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> > Note that sequence {F_(11,13,19)(n)} has period of length 9,
> > sequence {F_(13,19,23)(n)} has period of length 12,
> > sequence {F_(17,19,23,29)(n)} has period of length 15,
> > sequence {F_(19,23,31,53,59,89)(n)} has period of length 24,
> > while sequence {F_(23,29,73,233)(n)} has period of length 18, etc.
> 
> > ... periods are multiple of 3 (except for trivial case of {F_2(n)}
> > with period 1).
> 
>     That's because:
>     - if the set of divided-out primes has 2, the 
> period is obviously 1;
>     - if the set doesn't have 2, the divisions 
> don't affect the sequence
>       modulo 2, which remains 1, 1, 0, 
> (1, 1, 0)*. So if there's a period, 
>       it's a multiple of 3.
> 
> -- 
> Don Reble  djr at nk.ca
> 
> 
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 Shevelev Vladimir‎