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A056469 Number of elements in the continued fraction for Sum_{k=0..n} 1/2^2^k. 5
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Dec 07 2002
STATUS
approved

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Last modified April 16 23:37 EDT 2024. Contains 371756 sequences. (Running on oeis4.)