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Search: id:A059929
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| 0, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 20737, 54288, 142130, 372099, 974170, 2550408, 6677057, 17480760, 45765226, 119814915, 313679522, 821223648, 2149991425, 5628750624, 14736260450, 38580030723
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,500
M. Renault, Dissertation
M. Waldschmidt, Open Diophantine problems
E. H. Kuo, Applications of graphical condensation for enumerating matchings and tilings
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FORMULA
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a(n) = Fib(n+1)^2-(-1)^n = A007598(n+1)+A033999(n+1) = A000045(n+1)^2-A033999(n)
G.f.: [2x-x^2]/[(1+x)(1-3x+x^2)].
Sum[n=1..inf, 1/a(n)] = 1, Sum[n=1..inf, (-1)^n/a(n)] = 2-sqrt(5).
Sum[n=1..inf, 1/a(2n-1)] = 1/phi = (sqrt(5)-1)/2. - Franz Vrabec (franz.vrabec(AT)planetuniqa.at), Sep 15 2005
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
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MAPLE
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with (combinat):a:=n->fibonacci(n)*fibonacci(n+2): seq(a(n), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007
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PROGRAM
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(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]
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CROSSREFS
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Bisection of A070550.
First differences of A059840.
Adjacent sequences: A059926 A059927 A059928 this_sequence A059930 A059931 A059932
Sequence in context: A130002 A162034 A105286 this_sequence A123029 A103018 A005158
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Feb 09 2001
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