A056469 Number of elements in the continued fraction for Sum_{k=0..n} 1/2^2^k.

2, 3, 4, 6, 10, 18, 34, 66, 130, 258, 514, 1026, 2050, 4098, 8194, 16386, 32770, 65538, 131074, 262146, 524290, 1048578, 2097154, 4194306, 8388610, 16777218, 33554434, 67108866, 134217730, 268435458, 536870914, 1073741826, 2147483650

Offset: 0

Author

Benoit Cloitre, Dec 07 2002

Keywords

• nonn
• easy

Comments

Let f_1(x) := 1 - sqrt(1 - x^2) = 2*x^2 + 2*x^4 + 4*x^6 + ... and for n>1 let f_n(x) := f_{n-1}(f_1(x)) = x^(2^n)*(2 + 2^n*x^2 + 2^n*a(n-1)*x^4 + ...). - Michael Somos, Jun 29 2023

Examples

			G.f. = 2 + 3*x + 4*x^2 + 6*x^3 + 10*x^4 + 18*x^5 + 34*x^6 + ... - _Michael Somos_, Jun 29 2023
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A007400. Apart from initial term, same as A052548. See also A089985.

Programs

Magma

[Floor(2^(n-1)+2): n in [0..60]]; // Vincenzo Librandi, Sep 21 2011

Mathematica

LinearRecurrence[{3,-2},{2,3,4},40] (* Harvey P. Dale, Apr 23 2015 *)
a[ n_] := If[n < 0, 0, Floor[2^n/2] + 2]; (* Michael Somos, Jun 29 2023 *)

PARI

{a(n) = if(n<0, 0, 2^n\2 + 2)}; /* Michael Somos, Jun 29 2023 */

Sage

[floor(gaussian_binomial(n,1,2)+3) for n in range(-1,32)] # Zerinvary Lajos, May 31 2009

Formula

a(0)=2; for n > 0, a(n) = 2^(n-1) + 2 = A052548(n-1) + 2.

a(n) = floor(2^(n-1) + 2). - Vincenzo Librandi, Sep 21 2011

From Colin Barker, Mar 22 2013: (Start)

a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.

G.f.: -(x^2+3*x-2) / ((x-1)*(2*x-1)). (End)

E.g.f.: exp(x)*(2 + sinh(x)). - Stefano Spezia, Oct 19 2023