A392067 Decimal expansion of the limit mean of the number of continued fraction coefficients required per decimal digit for the golden ratio.

2, 3, 9, 2, 4, 8, 5, 9, 8, 3, 3, 9, 0, 8, 3, 2, 9, 8, 5, 6, 7, 9, 0, 9, 4, 8, 7, 6, 1, 8, 8, 3, 6, 8, 4, 5, 5, 1, 5, 7, 8, 0, 1, 4, 9, 1, 4, 8, 4, 1, 1, 6, 0, 4, 7, 0, 5, 2, 0, 9, 4, 1, 3, 9, 2, 6, 1, 5, 7, 9, 9, 4, 8, 4, 6, 7, 9, 9, 0, 6, 9, 5, 3, 9, 6, 9, 5

Offset

1,1

Comments

This limiting mean is the worst case possible.

The limiting mean of the continued fraction with some number k being infinitely repeated is given by 1/(2*log_10((k+sqrt(k^2+4))/2)).

The limiting mean of the continued fraction for almost all other real numbers is given by the Lochs's constant A086819.

Links

Table of n, a(n) for n=1..87.

Wikipedia, Lochs's theorem.

Formula

Equals 1/(2*log_10(phi)) = 1/log_10(1+phi).

Example

2.39248598339083298567909487618836845515...

Mathematica

N[1 / Log10[1 + GoldenRatio], 120]

Crossrefs

Cf. A001622, A086819.

Sequence in context: A030367 A028508 A021422 * A193086 A388231 A281289

Adjacent sequences: A392064 A392065 A392066 * A392068 A392069 A392070

Keyword

nonn,cons

Author

Jwalin Bhatt, Jan 27 2026

Status

approved