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Fibonacci sequence beginning 12, 67.
2

%I #35 Sep 08 2022 08:45:12

%S 12,67,79,146,225,371,596,967,1563,2530,4093,6623,10716,17339,28055,

%T 45394,73449,118843,192292,311135,503427,814562,1317989,2132551,

%U 3450540,5583091,9033631,14616722,23650353,38267075,61917428,100184503

%N Fibonacci sequence beginning 12, 67.

%C 20 of the first 100 terms are prime numbers.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

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

%F G.f.: (12*x+55*x^2)/(1-x-x^2). - _Philippe Deléham_, Nov 20 2008

%F a(n) = (61*sqrt(5)/5 + 6)*(1/2 + sqrt(5)/2)^n + (-61*sqrt(5)/5 + 6)*(1/2 - sqrt(5)/2)^n. - _Antonio Alberto Olivares_, May 23 2013

%e a(0) = (61*sqrt(5)/5 + 6)*(1/2 + sqrt(5)/2)^0 + (-61*sqrt(5)/5 + 6)*(1/2 - sqrt(5)/2)^0 =(61*sqrt(5)/5 + 6) + (-61*sqrt(5)/5 + 6) = 6 + 6 = 12.

%t a={};b=12;c=67;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* _Vladimir Joseph Stephan Orlovsky_, Jul 23 2008 *)

%t LinearRecurrence[{1,1},{12,67},50] (* or *) Rest[ CoefficientList[ Series[ (12x+55x^2)/(1-x-x^2),{x,0,50}],x]] (* _Harvey P. Dale_, May 11 2011 *)

%o (Magma) a0:=12; a1:=67; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..35]]; // _Vincenzo Librandi_, Nov 02 2014

%K nonn,easy

%O 1,1

%A _G. L. Honaker, Jr._, Dec 17 2003