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

0,1

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Index entries for linear recurrences with constant coefficients, signature (3,-2).

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

Example

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

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 *)

Prog

(Sage) [floor(gaussian_binomial(n, 1, 2)+3) for n in range(-1, 32)] # Zerinvary Lajos, May 31 2009
(Magma) [Floor(2^(n-1)+2): n in [0..60]]; // Vincenzo Librandi, Sep 21 2011
(PARI) {a(n) = if(n<0, 0, 2^n\2 + 2)}; /* Michael Somos, Jun 29 2023 */

Crossrefs

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

Sequence in context: A106511 A024490 A317200 * A228863 A004047 A355191

Adjacent sequences: A056466 A056467 A056468 * A056470 A056471 A056472

Keyword

nonn,easy

Author

Benoit Cloitre, Dec 07 2002

Status

approved