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A079004 Least x>=3 such that F(x)==1 (mod 3^n) where F(x) denotes the x-th Fibonacci number (A000045).

%I A079004 #22 Aug 16 2024 12:37:59
%S A079004 7,10,10,34,106,322,970,2914,8746,26242,78730,236194,708586,2125762,
%T A079004 6377290,19131874,57395626,172186882,516560650,1549681954,4649045866,
%U A079004 13947137602,41841412810,125524238434,376572715306,1129718145922
%N A079004 Least x>=3 such that F(x)==1 (mod 3^n) where F(x) denotes the x-th Fibonacci number (A000045).
%D A079004 R. L. Graham, D. E. Knuth and O. Patashnick, "Concrete Mathematics", second edition, Addison Wesley, ex. 6.59.
%H A079004 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A079004 a(1)=7, a(2)=10, a(3)=10; for n>3, a(n) = 3*a(n-1) + 4.
%F A079004 a(n) = 4*3^(n-2)-2 for n >= 3.
%F A079004 G.f.: 8*x^2+(23/3)*x+14/9+2/(x-1)-4/(9*(3*x-1)). - _Robert Israel_, Jan 15 2015
%p A079004 7, 10, seq(4*3^(n-2)-2,n=3..50); # _Robert Israel_, Jan 15 2015
%t A079004 a=2;lst={7,10};Do[a=a*3+4;AppendTo[lst,a],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 25 2008 *)
%t A079004 LinearRecurrence[{4,-3},{7,10,10,34},40] (* _Harvey P. Dale_, Aug 16 2024 *)
%o A079004 (PARI) a(n)=if(n<0,0,x=3; while((fibonacci(x)-1)%(3^n)>0,x++); x)
%Y A079004 Cf. A000045, A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774.
%K A079004 nonn,easy
%O A079004 1,1
%A A079004 _Benoit Cloitre_, Feb 01 2003
%E A079004 Formula corrected by _Robert Israel_, Jan 15 2015