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%I #33 Oct 19 2023 17:02:17
%S 2,3,4,6,10,18,34,66,130,258,514,1026,2050,4098,8194,16386,32770,
%T 65538,131074,262146,524290,1048578,2097154,4194306,8388610,16777218,
%U 33554434,67108866,134217730,268435458,536870914,1073741826,2147483650
%N Number of elements in the continued fraction for Sum_{k=0..n} 1/2^2^k.
%C 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
%H Vincenzo Librandi, <a href="/A056469/b056469.txt">Table of n, a(n) for n = 0..2000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F a(0)=2; for n > 0, a(n) = 2^(n-1) + 2 = A052548(n-1) + 2.
%F a(n) = floor(2^(n-1) + 2). - _Vincenzo Librandi_, Sep 21 2011
%F From _Colin Barker_, Mar 22 2013: (Start)
%F a(n) = 3*a(n-1) - 2*a(n-2) for n > 2.
%F G.f.: -(x^2+3*x-2) / ((x-1)*(2*x-1)). (End)
%F E.g.f.: exp(x)*(2 + sinh(x)). - _Stefano Spezia_, Oct 19 2023
%e 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
%t LinearRecurrence[{3,-2},{2,3,4},40] (* _Harvey P. Dale_, Apr 23 2015 *)
%t a[ n_] := If[n < 0, 0, Floor[2^n/2] + 2]; (* _Michael Somos_, Jun 29 2023 *)
%o (Sage) [floor(gaussian_binomial(n,1,2)+3) for n in range(-1,32)] # _Zerinvary Lajos_, May 31 2009
%o (Magma) [Floor(2^(n-1)+2): n in [0..60]]; // _Vincenzo Librandi_, Sep 21 2011
%o (PARI) {a(n) = if(n<0, 0, 2^n\2 + 2)}; /* _Michael Somos_, Jun 29 2023 */
%Y Cf. A007400. Apart from initial term, same as A052548. See also A089985.
%K nonn,easy
%O 0,1
%A _Benoit Cloitre_, Dec 07 2002
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