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A040037 Continued fraction for sqrt(44).

%I A040037 #35 Jul 27 2025 10:04:29
%S A040037 6,1,1,1,2,1,1,1,12,1,1,1,2,1,1,1,12,1,1,1,2,1,1,1,12,1,1,1,2,1,1,1,
%T A040037 12,1,1,1,2,1,1,1,12,1,1,1,2,1,1,1,12,1,1,1,2,1,1,1,12,1,1,1,2,1,1,1,
%U A040037 12,1,1,1,2,1,1,1,12,1,1,1,2,1
%N A040037 Continued fraction for sqrt(44).
%D A040037 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.
%H A040037 Harry J. Smith, <a href="/A040037/b040037.txt">Table of n, a(n) for n = 0..20000</a>
%H A040037 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.
%H A040037 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.
%H A040037 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,1).
%F A040037 From _Amiram Eldar_, Nov 12 2023: (Start)
%F A040037 Multiplicative with a(2) = 1, a(4) = 2, a(2^e) = 12 for e >= 3, and a(p^e) = 1 for an odd prime p.
%F A040037 Dirichlet g.f.: zeta(s) * (1 + 5/2^(3*s-1) + 1/4^s). (End)
%F A040037 G.f.: (6 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 6*x^8)/(1 - x^8). - _Stefano Spezia_, Jul 27 2025
%e A040037 6.633249580710799698229865473... = 6 + 1/(1 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, Jun 05 2009
%p A040037 Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
%t A040037 ContinuedFraction[Sqrt[44],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 06 2011 *)
%t A040037 PadRight[{6},80,{12,1,1,1,2,1,1,1}] (* _Harvey P. Dale_, Apr 02 2013 *)
%o A040037 (PARI) { allocatemem(932245000); default(realprecision, 14000); x=contfrac(sqrt(44)); for (n=0, 20000, write("b040037.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 05 2009
%Y A040037 Cf. A010498 (decimal expansion).
%K A040037 nonn,cofr,easy,mult
%O A040037 0,1
%A A040037 _N. J. A. Sloane_