A369346 Continued fraction expansion of the real root of x^3 - x^2 - 1 = 0.

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, 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, 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

Offset: 0

Author

Patrick McKinley, Jan 20 2024

Keywords

• nonn
• cofr

Links

• Patrick McKinley, Table of n, a(n) for n = 0..12174
• Eric Weisstein's World of Mathematics, Supergolden Ratio.

Crossrefs

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

Programs

Mathematica

ContinuedFraction[x/.First[Solve[x^3-x^2-1==0,x]],92] (* Stefano Spezia, Jan 21 2024 *)

PARI

\p100 \\ realprecision
contfrac(solve(x = 1, 2, x^3 - x^2 - 1),, 80) \\ Hugo Pfoertner, Jan 21 2024

bc

/* The "test" calculation evaluates the cubic to confirm the calculation of the root. */
define iter(frac)
{j = 0
 while(frac > 1){
   frac -= 1;
   j+=1}
 j
 return 1/frac}
scale=12578
f=(1+(e(l(((29+3*sqrt(93))/2))/3))+(e(l(((29-3*sqrt(93))/2))/3)))/3
psi=f
test=(psi-1)*psi*psi-1
for(i=0;i<12175;i++)f=iter(f)

Extensions

Offset changed by Andrew Howroyd, Feb 14 2025