A001601 a(n) = 2*a(n-1)^2 - 1, if n>1. a(0)=1, a(1)=3. (Formerly M3042 N1234)

1, 3, 17, 577, 665857, 886731088897, 1572584048032918633353217, 4946041176255201878775086487573351061418968498177

Offset

0,2

Comments

Reduced numerators of Newton's iteration for sqrt(2). - Eric W. Weisstein

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004

Evaluation of the 2^n - 1 degree interpolating polynomial of 1/x at Chebyshev nodes in the interval (1,2): v = 1.0; for(i = 1, n, v *= 4*(a(i) - x*v)); v *= 2/a(n+1). - Jose Hortal, Apr 07 2012

Smallest positive integer x satisfying the Pell equation x^2 - 2^(2*n+1) * y^2 = 1, for n > 0. - A.H.M. Smeets, Sep 29 2017

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. 376.

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).

Links

Dennis Martin, Table of n, a(n) for n = 0..11

Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy]

M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., 65 (1989), 213-220.

Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.

Eric Weisstein's World of Mathematics, Newtons Iteration.

Eric Weisstein's World of Mathematics, Square Root.

Eric Weisstein's World of Mathematics, Pythagoras's Constant.

H. S. Wilf, D. C. B. Marsh and J. V. Whittaker, Problem E1093, Amer. Math. Monthly, 61 (1954), 424-425.

Index entries for sequences related to Engel expansions

Formula

From Mario Catalani (mario.catalani(AT)unito.it), May 27 2003, May 30 2003: (Start)

a(n) = a(n-1)^2 + 2*A051009(n)^2 for n > 0.

a(n)^2 = 2*A051009(n+1)^2 + 1.

a(n) = Sum_{r=0..2^(n-1)} binomial(2^n, 2*r)*2^r. (End)

Expansion of 1/sqrt(2) as an infinite product: 1/sqrt(2) = Product_{k>=1} (1 - 1/(a(n)+1)). a(1)=3; a(n) = floor(1/(1-1/(sqrt(2)*Product_{k=1..n-1} 1-1/(a(k)+1)))). - Thomas Baruchel, Nov 06 2003

2*a(n+1) = A003423(n).

a(n) = (1/2)*((1 + sqrt(2))^(2^n) + (1 - sqrt(2))^(2^n)). - Artur Jasinski, Oct 10 2008

For n > 1: a(n) - 1 = 4^n * Product_{i=1..n-2} a(i)^2. - Jose Hortal, Apr 13 2012

From Peter Bala, Nov 11 2012: (Start)

4*sqrt(2)/7 = Product_{n>=1} (1 - 1/(2*a(n))).

sqrt(2) = Product_{n>=1} (1 + 1/a(n)).

a(n) = (1/2)*A003423(n-1). (End)

a(n) = cos(2^(n-1) * arccos(3)) = cosh(2^(n-1) * log(3 + 2*sqrt(2))) for n >= 1. - Daniel Suteu, Jul 28 2016

a(n+1) = T(2^n,3), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Feb 01 2017

a(n) = A001541(2^(n-1)). - A.H.M. Smeets, May 28 2017

Mathematica

Table[Simplify[Expand[(1/2) ((1 + Sqrt[2])^(2^n) + (1 - Sqrt[2])^(2^n))]], {n, 0, 7}]  (* Artur Jasinski, Oct 10 2008 *)
Join[{1}, NestList[2#^2-1&, 3, 7]]  (* Harvey P. Dale, Mar 24 2011 *)

Prog

(PARI) a(n)=if(n<1, n==0, 2*a(n-1)^2-1)

Crossrefs

Cf. A051009, A003423.

a(n) = A001333(2^n).

Sequence in context: A305375 A128300 A292082 * A061119 A094133 A049985

Adjacent sequences: A001598 A001599 A001600 * A001602 A001603 A001604

Keyword

nonn,easy,nice,frac

Author

N. J. A. Sloane

Status

approved