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A002812
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a(n) = 2*a(n-1)^2 - 1, starting a(0)=2.
(Formerly M1817 N0720)
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4
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2, 7, 97, 18817, 708158977, 1002978273411373057, 2011930833870518011412817828051050497, 8095731360557835890888779535060256832479295062749579257164654370487894017
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| An infinite coprime sequence defined by recursion - Michael Somos, Mar 14 2004
2^p-1 is prime iff it divides a(p-2), since a(n) = A003010(n)/2, where A003010 is the Lucas-Lehmer sequence used for Mersenne number primality testing - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Mar 09 2007
Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 2008: (Start)
Also numerators of the convergents to the square root of 3 using the
the following recursion for initial x = 1.
x1=x
x=3/x
x=(x+x1)/2
This recursion was derived by experimenting with polynomial recursions of the
form x = -a(0)/(a(n-1)x^(n-1)+...+a(1)) in an effort to find a root for the
polynomial a(n)x^n+a(n-1)x^(n-1)+...+a(0). The process was hit and miss until
I introduced the averaging step described above. This method is equivelant to
Newton's Method although derived somewhat differently. (End)
The sequence satisfies the Pell equation a(n)^2 - 3*A071579(n)^2 = 1. - Vincenzo Librandi, Dec 19 2011
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REFERENCES
| L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 399.
E. Lucas, Nouveaux theoremes d'arithmetique superieure, Comptes Rend., 83 (1876), 1286-1288.
M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., 65 (1989), 213-220.
J. O. Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10
J. O. Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
Eric Weisstein's World of Mathematics, Newton's Iteration
Index entries for sequences related to Engel expansions
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FORMULA
| a(n) = A001075(2^n).
a(n) = ((2+sqrt(3))^(2^n)+(2-sqrt(3))^(2^n))/2. - Bruno Berselli, Dec 20 2011
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MATHEMATICA
| Table[((2 + Sqrt[3])^2^n + (2 - Sqrt[3])^2^n)/2, {n, 0, 7}] (* Bruno Berselli, Dec 20 2011 *)
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PROG
| (PARI) a(n)=if(n<1, 2*(n==0), 2*a(n-1)^2-1)
Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 2008: (Start)
(PARI) /* Roots by recursion. Find first root of ax^2 + b^x + c */
rroot2(a, b, c, p) =
{
local(x=1, x1=1, j);
for(j=1, p,
x1=x;
x=-c/(a*x+b);
x=(x1+x)/2; /* Let x be the average of the last 2 values */
print1(numerator(x)", ");
);
}
(End)
(MAGMA) I:=[2]; [n le 1 select I[n] else 2*Self(n-1)^2-1: n in [1..8]]; // Vincenzo Librandi, Dec 19 2011
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CROSSREFS
| Cf. A001075, A003010, A071579.
Sequence in context: A072059 A102344 A087589 * A192342 A102598 A206157
Adjacent sequences: A002809 A002810 A002811 * A002813 A002814 A002815
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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