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Search: id:A084765
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| A084765 |
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a(n)=2a(n-1)^2-1, a(0)=1, a(1)=5. |
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+0
1
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| 1, 5, 49, 4801, 46099201, 4250272665676801, 36129635465198759610694779187201, 2610701117696295981568349760414651575095962187244375364404428801
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Product((1+1/a(k)), k=1,..,n) converges to sqrt(3/2).
A subsequence of A001079 (cf. formula), which must contain any prime occurring in A001079. The initial term a(0)=1 seems rather unnatural; using the recurrence relation it would yield the constant sequence 1,1,1,... Note that this sequence corresponds to sequence b(n) in Shallit's paper, which starts only at offset n=1. [From M. F. Hasler (MHasler(AT)univ-ag.fr), Sep 27 2009]
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REFERENCES
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H. S. Wilf, Limit of a sequence, Elementary Problem E 1093, Amer. Math. Monthly 61 (1954), 424-425
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LINKS
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J. O. Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
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FORMULA
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With a=5+2sqrt(6), b=5-2sqrt(6): a(n+1)=(a^(2^n)+b^(2^n))/2.
A084765(n+1)=A001079(2^n). [From M. F. Hasler (MHasler(AT)univ-ag.fr), Sep 27 2009]
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MATHEMATICA
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For n>0: b[n_] := b[n] = 2 b[n - 1]^2 - 1; b[1] = 5 Table[b[n], {n, 1, 8}]
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CROSSREFS
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Cf. A084764.
Sequence in context: A001819 A064618 A075986 this_sequence A082795 A059008 A136890
Adjacent sequences: A084762 A084763 A084764 this_sequence A084766 A084767 A084768
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Jun 04 2003
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