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

Vladimir Shevelev shevelev at bgu.ac.il
Mon Apr 1 19:32:59 CEST 2013

```Dear SeqFans,

For a given prime p, consider Fbonacci-like numbers {F_p(n)} without moltiples of p which defined in the following way:
a(0)=0, a(1)=1, for n>=2, a(n)=a(n-1)+a(n-2), if a(n-1)+a(n-2) is not multiple of p, otherwise, a(n)=a(n-1)+a(n-2) divided by maximal possible power of p.
Note that {F_2(n)} has period of length 1, {F_3(n)} has period of length 3, {F_5(n)} has period of length 6. The first case when such a sequence has not a trivial period and, probably, is non-periodic, is p=7 (A224196). I do not know, if this sequence is bounded?
Other questions arise when we consider Fibonacci-like sequences without multiples of several primes which defined quite analogously: e.g., for {F_(p,q)(n)}, a(0)=0, a(1)=1, for n>=2, a(n)=a(n-1)+a(n-2), if a(n-1)+a(n-2) is multiple of neither p no q, otherwise, a(n)=a(n-1)+a(n-2) divided by maximal possible power of p and maximal possible power of q. The first question is the following: for a given prime p, whether exist one or more primes greater than p, such that Fibonacci-like sequences without multiples of these (together with p) primes is periodic? So in case p=7 we could add primes 11 and 13 and obtain an eventually periodic sequence {F_(7,11,13)(n)} with period of length 12: 0,1,1,2,3,5,8,1,9,10,19,29,48,1,1,2,3,5,...
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.
By another strange observation, all lengths of periods are multiple of 3 (except for trivial case of {F_2(n)} with period 1). Could anyone to find a periodic Fibonacci-like sequences without multiples of several primes with a length of period not multiple of 3?

Best regards,