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Continued fraction for sqrt(257).
4

%I #26 Feb 14 2024 10:46:20

%S 16,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,

%T 32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,

%U 32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32,32

%N Continued fraction for sqrt(257).

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F From _Elmo R. Oliveira_, Feb 13 2024: (Start)

%F a(n) = 32 for n >= 1.

%F G.f.: 16*(1+x)/(1-x).

%F E.g.f.: 32*exp(x) - 16.

%F a(n) = 16*A040000(n) = 8*A040002(n) = 4*A040012(n) = 2*A040056(n). (End)

%e 16 + 1/(32 + 1/(32 + 1/(32 + 1/(32 + ...)))) = sqrt(257).

%p with(numtheory): Digits := 300: convert(evalf(sqrt(257)),confrac);

%t Block[{$MaxExtraPrecision=1000}, ContinuedFraction[Sqrt[257],100]] (* or *) PadRight[{16},100,{32}] (* _Harvey P. Dale_, Aug 15 2021 *)

%o (PARI) a(n)=if(n,32,16) \\ _Charles R Greathouse IV_, Apr 08 2012

%Y Cf. A041480/A041481 (convergents).

%Y Cf. A040000, A040002, A040012, A040056.

%K nonn,cofr,easy

%O 0,1

%A _N. J. A. Sloane_