0,1

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)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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

(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)

Cf. A071579.

a(n) = A001075(2^n).

Cf. A003010.

Sequence in context: A072059 A102344 A087589 this_sequence A102598 A102747 A122524

Adjacent sequences: A002809 A002810 A002811 this_sequence A002813 A002814 A002815

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).