Search: id:A059929
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%I A059929
%S A059929 0,2,3,10,24,65,168,442,1155,3026,7920,20737,54288,142130,372099,
%T A059929 974170,2550408,6677057,17480760,45765226,119814915,313679522,
%U A059929 821223648,2149991425,5628750624,14736260450,38580030723
%N A059929 Fib(n)*Fib(n+2).
%C A059929 Expansion of golden ratio (1+sqrt(5))/2 as an infinite product: phi = 
               prod(i=0, infty, (1+1/(fibonacci(2i+1) * fibonacci(2i+3)-1)) * (1-1/
               (fibonacci(2i+2) * fibonacci(2i+4)+1))) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), 
               Nov 11 2003
%H A059929 Harry J. Smith, <a href="b059929.txt">Table of n, a(n) for n=0,...,500</
               a>
%H A059929 M. Renault, <a href="http://www.math.temple.edu/~renault/fibonacci/thesis.html">
               Dissertation</a>
%H A059929 M. Waldschmidt, <a href="http://arXiv.org/abs/math.NT/0312440">Open Diophantine 
               problems</a>
%H A059929 E. H. Kuo, <a href="http://arXiv.org/abs/math.CO/0304090">Applications 
               of graphical condensation for enumerating matchings and tilings</
               a>
%F A059929 a(n) = Fib(n+1)^2-(-1)^n = A007598(n+1)+A033999(n+1) = A000045(n+1)^2-A033999(n)
%F A059929 G.f.: [2x-x^2]/[(1+x)(1-3x+x^2)].
%F A059929 Sum[n=1..inf, 1/a(n)] = 1, Sum[n=1..inf, (-1)^n/a(n)] = 2-sqrt(5).
%F A059929 Sum[n=1..inf, 1/a(2n-1)] = 1/phi = (sqrt(5)-1)/2. - Franz Vrabec (franz.vrabec(AT)aon.at), 
               Sep 15 2005
%F A059929 1 = 1/2 + 1/3 + 1/10 + 1/24 + 1/65 + 1/168 + ..., = 1/(1*2) + 1/(1*3) 
               + 1/(2*5) + 1/(3*8) + ... - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Mar 16 2008
%F A059929 Sum[n=1..inf, 1/a(2n)] = (3-sqrt(5))/2. [From Franz Vrabec (franz.vrabec(AT)aon.at), 
               Nov 30 2009]
%p A059929 with (combinat):a:=n->fibonacci(n)*fibonacci(n+2): seq(a(n), n=0..26); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007
%t A059929 Table[Fibonacci[n]*Fibonacci[n+2],{n,0,60}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Nov 17 2009]
%o A059929 (PARI) { for (n=0, 500, write("b059929.txt", n, " ", fibonacci(n)*fibonacci(n 
               + 2)); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 
               30 2009]
%Y A059929 Bisection of A070550.
%Y A059929 First differences of A059840.
%Y A059929 Sequence in context: A130002 A162034 A105286 this_sequence A123029 A103018 
               A005158
%Y A059929 Adjacent sequences: A059926 A059927 A059928 this_sequence A059930 A059931 
               A059932
%K A059929 nonn,new
%O A059929 0,2
%A A059929 Henry Bottomley (se16(AT)btinternet.com), Feb 09 2001

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