%I A063896
%S A063896 0,1,1,3,7,31,255,8191,2097151,17179869183,36028797018963967,
%T A063896 618970019642690137449562111,
%U A063896 22300745198530623141535718272648361505980415
%N A063896 2^Fibonacci(n) - 1.
%C A063896 The recurrence can also be written a(n)+1=(a(n-1)+1)*(a(n-2)+1) or log_p(a(n)+1)=log_p(a(n-1)+1)+log_p(a(n-2)\
               +1), respectively. Setting a(1)=p-1 for any natural p>1, it follows 
               that log_p(a(n)+1)=Fib(n). Hence any other sequence p^Fib(n)-1 could 
               also serve as a valid solution to that recurrence, only depending 
               on the value of the term a(1). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 27 2007
%C A063896 Written in binary, a(n) contains Fib(n) 1's (Fib(n)=A000045(n)). Thus 
               the sequence converted to base-2 is A007088(a(n))=0,1,1,11,111,11111,
               11111111,... - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 
               Jun 27 2007
%F A063896 The solution to the recurrence a(0) = 0; a(1) = 1; a(n) = a(n-1)*a(n-2) 
               + a(n-1) + a(n-2).
%F A063896 a(n)=A000301(n+1)-1. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 
               26 2007
%F A063896 a(n)=a(n-2)*2^ceiling(log_2(a(n-1)))+a(n-1) for n>1. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Jun 27 2007
%t A063896 a[0, k_] = 0; a[1, k_] = 1; a[n_, k_] := (k - 1)*a[n - 1, k]*a[n - 2, 
               k] + a[n - 1, k] + a[n - 2, k]; Table[ a[n, 2], {n, 0, 14} ]
%t A063896 a=0;b=1;lst={a,b};Do[c=a*b+a+b;AppendTo[lst,c];a=b;b=c,{n,3*3!}];lst 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 13 2009]
%Y A063896 Cf. A000045 and A000301.
%Y A063896 Cf. A000045, A061107.
%Y A063896 See A131293 for a base-10 analog with Fib(n) 1's.
%Y A063896 Sequence in context: A073917 A030521 A105767 this_sequence A156895 A074047 
               A121810
%Y A063896 Adjacent sequences: A063893 A063894 A063895 this_sequence A063897 A063898 
               A063899
%K A063896 nonn
%O A063896 0,4
%A A063896 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 29 2001