OFFSET
0,2
COMMENTS
Also the denominators of the convergents to sqrt(3) using Newton's recursion x = (3/x+x)/2. - Cino Hilliard, Sep 28 2008
For n>1, Egyptian fraction of 2-sqrt(3): 2-sqrt(3) = 1/4 + 1/56 + 1/10864 + 1/408855776 + ... - Simon Plouffe, Feb 20 2011
The sequence satisfies the Pell equation A002812(n)^2-3*a(n)^2 = 1. - Vincenzo Librandi, Dec 19 2011
From Peter Bala, Oct 30 2013: (Start)
Apart from giving the numerators in the Engel series representation of 2 - sqrt(3), as stated above by Plouffe, this sequence is also a Pierce expansion of the real number x = 2 - sqrt(3) to the base b := 1/sqrt(12) (see A058635 for a definition of this term).
The associated series representation begins 2 - sqrt(3) = b/1 - b^2/(1*4) + b^3/(1*4*56) - b^4/(1*4*56*10864) + .... Cf. A230338.
More generally, for n >= 0, the sequence [a(n), a(n+1), a(n+2), ...] gives a Pierce expansion of (2 - sqrt(3))^(2^n) to the base b = 1/sqrt(12). Some examples are given below. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10
Eric Weisstein's World of Mathematics, Newton's Iteration.
Doron Zeilberger, Another Book of Somos-Like Miracles, Prop. Number, 2; Local copy.
FORMULA
a(n) = 1/sqrt(12)*( (2 + sqrt(3))^2^n - (2 - sqrt(3))^2^n ) = A001353(2^n).
a(n) = 2*a(n-1)*(6*a(n-2)^2+1). - Max Alekseyev, Apr 19 2006
Recurrence equations:
a(n)/a(n-1) = (a(n-1)/a(n-2))^2 - 2 for n >= 2.
a(n) = a(n-1)*sqrt(12*a(n-1)^2 + 4) for n >= 1. - Peter Bala, Oct 30 2013
0 = 6*a(n)^2*a(n+2) - 6*a(n+1)^3 - 2*a(n+1) + a(n+2) for n>=1. - Michael Somos, Dec 05 2016
0 = a(n)^2*(2*a(n+1) + a(n+2)) - a(n+1)^3 for n>=1. - Michael Somos, Dec 05 2016
a(n) = A001353(2^n). - Michael Somos, Jul 29 2024
EXAMPLE
Let b = 1/sqrt(12) and x = 2 - sqrt(3). We have the following Pierce expansions to base b:
x = b/1 - b^2/(1*4) + b^3/(1*4*56) - b^4/(1*4*56*10864) + b^5/(1*4*56*10864*408855776) - ....
x^2 = b/4 - b^2/(4*56) + b^3/(4*56*10864) - b^4/(4*56*10864*408855776) + ....
x^4 = b/56 - b^2/(56*10864) + b^3/(56*10864*408855776) - ....
x^8 = b/10864 - b^2/(10864*408855776) + .... - Peter Bala, Oct 30 2013
MATHEMATICA
a[ n_] := If[n<0, 0, Coefficient[PolynomialMod[x^2^n, x^2 - 4*x + 1], x]]; (* Michael Somos, Jul 29 2024 *)
a[ n_] := If[n<1, Boole[n==0], a[n] = a[n-1]*Sqrt[12*a[n-1]^2 + 4] ]; (* Michael Somos, Jul 29 2024 *)
PROG
(PARI) g(n, p) = x=1; for(j=1, p, x=(n/x+x)/2; print1(denominator(x)", "))
(PARI) {a(n) = if(n<0, 0, imag((2 + quadgen(12))^2^n))}; /* Michael Somos, Jul 29 2024 */
(PARI) {a(n) = if(n<0, 0, polcoef(lift(Mod(x^2^n, x^2 - 4*x + 1)), 1))}; /* Michael Somos, Jul 29 2024 */
g(3, 8) \\ Cino Hilliard, Sep 28 2008
(Magma) I:=[1, 4]; [n le 2 select I[n] else 2*Self(n-1)*(6*Self(n-2)^2+1): n in [1..8]]; // Vincenzo Librandi, Dec 19 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Joe Keane (jgk(AT)jgk.org), May 31 2002
STATUS
approved