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A084765
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a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=5.
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5
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OFFSET
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0,2
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COMMENTS
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Product_{k>=1} (1 + 1/a(k)) = sqrt(3/2) (see A010527).
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. - M. F. Hasler, Sep 27 2009
Since if x is even (x^2-2)/2 = 2*y^2-1 and 10 is even from a(1) onward this is a reduced version of the LL sequence starting with 10 (A135927) as it is reduced by dividing by 2 it is also the difference between two possible LL sequences. - Roderick MacPhee, May 31 2015
For n >= 3, a(n) == 201 (mod 1000) if n is even, a(n) == 801 (mod 1000) if n is odd. - Robert Israel, Jun 01 2015
The next term -- a(8) -- has 128 digits. - Harvey P. Dale, Mar 28 2020
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LINKS
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Table of n, a(n) for n=0..7.
Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
H. S. Wilf, Limit of a sequence, Elementary Problem E 1093, Amer. Math. Monthly 61 (1954), 424-425.
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FORMULA
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a(n+1) = (x^(2^n) + y^(2^n))/2, with x = 5 + 2*sqrt(6), y = 5 - 2*sqrt(6).
a(n+1) = A001079(2^n). - M. F. Hasler, Sep 27 2009
4*sqrt(6)/11 = Product_{n >= 1} (1 - 1/(2*a(n))). See A002812 for some general properties of the recurrence a(n+1) = 2*a(n)^2 - 1. - Peter Bala, Nov 11 2012
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MAPLE
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1, seq(expand((5+2*sqrt(6))^(2^n)+(5-2*sqrt(6))^(2^n))/2, n=0..10); # Robert Israel, Jun 01 2015
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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}]
Join[{1}, NestList[2 #^2 - 1 &, 5, 10]] (* Harvey P. Dale, Mar 28 2020 *)
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PROG
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(MAGMA) I:=[1, 5]; [n le 2 select I[n] else 2*Self(n-1)^2-1: n in [1..10]]; // Vincenzo Librandi, Jun 02 2015
(PARI) first(m)={my(v=[1, 5]); for(i=3, m, v=concat(v, 2*v[i-1]^2 - 1)); v; } \\ Anders Hellström, Aug 22 2015
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CROSSREFS
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Cf. A084764, A002812.
Sequence in context: A224680 A075986 A251657 * A203411 A218322 A247707
Adjacent sequences: A084762 A084763 A084764 * 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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STATUS
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approved
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