%I #22 Nov 16 2023 15:55:34

%S 1,3,2,6,12,6,16,12,6,12,8,6,52,48,12,24,16,6,40,12,16,24,22,12,60,

%T 156,18,48,28,12,64,48,8,48,48,6,76,120,52,12,28,48,42,24,12,66,96,24,

%U 112,60,16,156,26,18,24,48,40,84,24,12,30,192,48,96,156,24,136,48,22,48,144

%N Pisano period of the 3-Fibonacci numbers A006190.

%C Period of the sequence defined by reading A006190 modulo n.

%H Vincenzo Librandi, <a href="/A175182/b175182.txt">Table of n, a(n) for n = 1..1000</a>

%H Sergio Falcon and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2008.02.014">k-Fibonacci sequences modulo m</a>, Chaos, Solit. Fractals 41 (2009), 497-504.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PisanoPeriod.html">Pisano period</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pisano_period">Pisano period</a>.

%p F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc:

%p Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc:

%p k := 3 ; seq( Pper(k,m),m=1..80) ;

%t Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[3*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* _Vincenzo Librandi_, Dec 20 2012, _T. D. Noe_ *)

%Y Cf. A001175, A006190, A175181, A175183, A175184, A175185.

%K nonn,easy

%O 1,2

%A _R. J. Mathar_, Mar 01 2010