A393413 Decimal expansion of the limit mean of the number of continued fraction coefficients required per decimal digit for the bronze ratio.

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

Offset

0,1

Comments

For this limit, the number of decimal digits k determined by n continued fraction terms is defined by the magnitude of the approximation error: k is such that the absolute difference between the n-th convergent and the true value lies between 10^-(k+1) and 10^-k.

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)).

This constant is given by the smallest value of k for which the resulting mean is <1.

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

It is also the constant given by the value of k for which the resulting mean is the closest to the Lochs's constant.

Links

Table of n, a(n) for n=0..86.

Wikipedia, Lochs's theorem.

Formula

Equals 1/(2*log_10((3+sqrt(13))/2)) = 1/log_10((11+3*sqrt(13))/2).

Example

0.963615660273846395420468439246971336489...

Mathematica

N[1 / Log10[(11 + 3Sqrt[13]) / 2], 120]

Crossrefs

Cf. A098316, A086819, A392067, A393243.

Sequence in context: A271172 A019683 A020759 * A226582 A276538 A243313

Adjacent sequences: A393410 A393411 A393412 * A393414 A393415 A393416

Keyword

nonn,cons

Author

Jwalin Bhatt, Feb 14 2026

Status

approved