[seqfan] Re: Periodic Fibonacci-like sequences without multiples of several primes
Don Reble
djr at nk.ca
Thu Apr 4 07:48:22 CEST 2013
> 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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