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A010127 Continued fraction for sqrt(23).

%I A010127 #45 Jul 26 2025 12:49:12
%S A010127 4,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,
%T A010127 3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,
%U A010127 8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8,1,3,1,8
%N A010127 Continued fraction for sqrt(23).
%D A010127 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 95 at p. 262.
%D A010127 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 275-276.
%H A010127 Harry J. Smith, <a href="/A010127/b010127.txt">Table of n, a(n) for n = 0..20000</a>
%H A010127 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.
%H A010127 <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.
%H A010127 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).
%F A010127 From _Amiram Eldar_, Nov 12 2023: (Start)
%F A010127 Multiplicative with a(2) = 3, a(2^e) = 8 for e >= 2, and a(p^e) = 1 for an odd prime p.
%F A010127 Dirichlet g.f.: zeta(s) * (1 + 1/2^(s-1) + 5/4^s). (End)
%F A010127 From _Stefano Spezia_, Aug 17 2024: (Start)
%F A010127 G.f.: (4 + x + 3*x^2 + x^3 + 4*x^4)/(1 - x^4).
%F A010127 E.g.f.: (5*cos(x) + 11*cosh(x) + 2*sinh(x) - 8)/2. (End)
%e A010127 4.795831523312719541597438064... = 4 + 1/(1 + 1/(3 + 1/(1 + 1/(8 + ...)))). - _Harry J. Smith_, Jun 03 2009
%t A010127 ContinuedFraction[Sqrt[23],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 05 2011 *)
%t A010127 PadRight[{4},120,{8,1,3,1}] (* _Harvey P. Dale_, Oct 23 2024 *)
%o A010127 (PARI) { allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(23)); for (n=0, 20000, write("b010127.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 03 2009
%Y A010127 Cf. A010479 (decimal expansion).
%K A010127 nonn,cofr,easy,mult
%O A010127 0,1
%A A010127 _N. J. A. Sloane_