A328912 Continued fraction expansion of log_2((sqrt(5)+1)/2) = 0.6942419... = A242208.

0, 1, 2, 3, 1, 2, 3, 2, 4, 2, 1, 2, 11, 2, 1, 11, 1, 1, 134, 2, 2, 2, 1, 4, 1, 1, 3, 1, 7, 1, 13, 1, 3, 5, 1, 1, 1, 8, 1, 3, 4, 1, 1, 1, 3, 4, 1, 3, 1, 4, 1, 4, 1, 3, 40, 1, 1, 5, 4, 3, 3, 1, 3, 1, 2, 6, 1, 1, 2, 28, 11, 1, 71, 2, 1, 4, 8, 5, 1, 2, 1, 1, 14

Offset: 0

Author

M. F. Hasler, Oct 31 2019

Keywords

• nonn
• cofr

Comments

This number is also the solution to 1 + 2^x = 4^x, or 1 + 1/2^x = 2^x, which clarifies the relation to Phi = (sqrt(5)+1)/2, solution to 1 + 1/x = x.

Examples

			log_2((sqrt(5)+1)/2) = 0.6942419... = 0 + 1/(1 + 1/(2 + 1/(3 + 1/(1 + ...))))
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A242208, A001622 (decimals of Phi), A000012 (cont. frac. of Phi).

Programs

Mathematica

ContinuedFraction[Log2[GoldenRatio], 100] (* Paolo Xausa, Mar 07 2024 *)

PARI

localprec(1000); contfrac(log(sqrt(5)+1)/log(2)-1)

Extensions

Some terms corrected by Paolo Xausa, Mar 07 2024