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