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A078414 a(n) = (a(n-1)+a(n-2))/7^k, where 7^k is the highest power of 7 dividing a(n-1)+a(n-2).

%I A078414 #39 Mar 13 2024 08:35:37
%S A078414 1,1,2,3,5,8,13,3,16,19,5,24,29,53,82,135,31,166,197,363,80,443,523,
%T A078414 138,661,799,1460,2259,3719,122,3841,3963,7804,1681,1355,3036,4391,
%U A078414 1061,5452,6513,11965,18478,4349,3261,7610,1553,187,1740,1927,3667,5594,27
%N A078414 a(n) = (a(n-1)+a(n-2))/7^k, where 7^k is the highest power of 7 dividing a(n-1)+a(n-2).
%C A078414 From _Vladimir Shevelev_, Apr 01 2013; edited by _Danny Rorabaugh_, Feb 19 2016: (Start)
%C A078414 If we consider Fibonacci-like numbers {F_p(n)} without positive multiples of p, where p is a fixed prime, then {F_2(n)} has period of length 1, {F_3(n)} has period of length 3, {F_5(n)} has period of length 6. This sequence is the first which does not have a trivial period and, probably, even is non-periodic.
%C A078414 An open question: Is this sequence bounded?
%C A078414 Consider Fibonacci-like sequences without multiples of several primes, defined 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) divided by the maximal possible powers of p and q.
%C A078414 Problem: For what sets of primes is the corresponding Fibonacci-like sequence without multiples of these primes periodic?
%C A078414 Examples: sequence {F_(7,11,13)(n)} has period of length 12: 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 19, 29, 48, 1, 1, 2, 3, 5,...; {F_(11,13,19)(n)} has period of length 9; {F_(13,19,23)(n)} has period of length 12; {F_(17,19,23,29)(n)} has period of length 15; {F_(19,23,31,53,59,89)(n)} has period of length 24; {F_(23,29,73,233)(n)} has period of length 18.
%C A078414 _Don Reble_ noted that lengths of all such periods could only be multiples of 3 because every Fibonacci-like sequence considered here modulo 2 has the form 0,1,1,0,1,1,... .
%C A078414 (End)
%H A078414 Alois P. Heinz, <a href="/A078414/b078414.txt">Table of n, a(n) for n = 1..4000</a>
%H A078414 B. Avila, T. Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>.
%F A078414 a(n) = A242603(a(n-1)+a(n-2)). - _R. J. Mathar_, Mar 13 2024
%p A078414 a:= proc(n) option remember; local t, j;
%p A078414       if n<3 then 1
%p A078414     else t:= a(n-1)+a(n-2);
%p A078414          while irem(t, 7, 'j')=0 do t:=j od; t
%p A078414       fi
%p A078414     end:
%p A078414 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 25 2012
%t A078414 nxt[{a_,b_}]:=Module[{n=IntegerExponent[a+b,7]},{b,(a+b)/7^n}]; Transpose[ NestList[nxt,{1,1},60]][[1]] (* _Harvey P. Dale_, Jul 23 2012 *)
%Y A078414 Cf. A000045, A078412, A214094, A214156, A216231, A216275, A216835.
%K A078414 nonn,easy
%O A078414 1,3
%A A078414 _Yasutoshi Kohmoto_, Dec 28 2002
%E A078414 Corrected by _Harvey P. Dale_, Jul 23 2012