A177412 - OEIS

%I #21 Oct 06 2021 20:39:27

%S 14831,41069,55900,96969,152869,249838,402707,652545,1055252,1707797,

%T 2763049,4470846,7233895,11704741,18938636,30643377,49582013,80225390,

%U 129807403,210032793,339840196,549872989,889713185,1439586174,2329299359,3768885533,6098184892,9867070425,15965255317,25832325742,41797581059

%N Fibonacci sequence beginning 14831,41069.

%C a(0)=14831 is a prime; the next prime number in the sequence is a(18604) = 2278143...6069, which has 3893 digits. (The initial values are chosen for this particularly long chain of consecutive composites.)

%H Colin Barker, <a href="/A177412/b177412.txt">Table of n, a(n) for n = 0..1000</a>

%H D. E. Knuth, <a href="http://www.jstor.org/stable/2691504">A Fibonacci-like sequence of composite numbers</a>, Math. Mag. 63 (1) (1990) 21-25.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

%F From _R. J. Mathar_, Dec 12 2010: (Start)

%F a(n) = a(n-1) + a(n-2).

%F G.f.: (14831+26238*x) / (1-x-x^2).

%F (End)

%F a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-67307+14831*sqrt(5)) + (1+sqrt(5))^n*(67307+14831*sqrt(5)))) / sqrt(5). - _Colin Barker_, May 03 2017

%p a:= n-> (<<0|1>, <1|1>>^n. <<14831, 41069>>)[1, 1]:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 06 2021

%t q=2;Do[Do[If[GCD[x,y]!=1,Break[]];a=x;b=y;lst={a,b};k=2;Do[If[PrimeQ[c=a+b],Break[]];k++;AppendTo[lst,c];a=b;b=c,{n,10!}];If[k>q,q=k;Print[If[Length[lst]>9,Take[lst,9],lst],k,"=",c]],{y,2*x+1,4*x+1}],{x,0,10!}]

%o (PARI) Vec((14831+26238*x) / (1-x-x^2) + O(x^30)) \\ _Colin Barker_, May 03 2017

%Y Cf. A000045.

%K nonn,easy

%O 0,1

%A _Vladimir Joseph Stephan Orlovsky_, Dec 10 2010