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Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.
1

%I #35 Feb 14 2025 17:27:04

%S 1,2,6,1,3,5,4,22,1,1,4,1,2,84,1,3,1,6,1,3,1,9,1,1,1,1,19,3,1,2,1,5,1,

%T 5,2,2,1,1,1,1,76,6,8,1,1,5,1,5,1,1,25,1,2,1,116,2,1,8,1,1,3,1,53,5,

%U 276,2,1,1,1,3,3,2,1,1,4,13,1,1,1,4,1,1,1,9,9,1,1,9,6,1,2,32

%N Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.

%H Patrick McKinley, <a href="/A369346/b369346.txt">Table of n, a(n) for n = 0..12174</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SupergoldenRatio.html">Supergolden Ratio</a>.

%t ContinuedFraction[x/.First[Solve[x^3-x^2-1==0,x]],92] (* _Stefano Spezia_, Jan 21 2024 *)

%o (bc)

%o /* The "test" calculation evaluates the cubic to confirm the calculation of the root. */

%o define iter(frac)

%o {j = 0

%o while(frac > 1){

%o frac -= 1;

%o j+=1}

%o j

%o return 1/frac}

%o scale=12578

%o f=(1+(e(l(((29+3*sqrt(93))/2))/3))+(e(l(((29-3*sqrt(93))/2))/3)))/3

%o psi=f

%o test=(psi-1)*psi*psi-1

%o for(i=0;i<12175;i++)f=iter(f)

%o (PARI)

%o \p100 \\ realprecision

%o contfrac(solve(x = 1, 2, x^3 - x^2 - 1),, 80) \\ _Hugo Pfoertner_, Jan 21 2024

%Y Cf. A092526 (decimal expansion), A381124, A381125 (convergents).

%K nonn,cofr,changed

%O 0,2

%A _Patrick McKinley_, Jan 20 2024

%E Offset changed by _Andrew Howroyd_, Feb 14 2025