A394765 Decimal expansion of the number defined by the continued fraction 1/(r+1/(r+1/(r + ...))), where r is the silver ratio (A014176).

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

Offset

0,1

Comments

[0; 1+sqrt(2), 1+sqrt(2), 1+sqrt(2), ...].

Repeat s = s + 1+sqrt(2); s=1/s. The initial value of s is irrelevant, as long as s != -(1+sqrt(2)).

An algebraic integer.

In general, for the metallic number M = [N; N, N, N, ...], the number defined by the continued fraction [0; M, M, M, ...] is the smallest positive real root of x^4 + N*x^3 - 3*x^2 - N*x + 1 = 0.

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..10000

Index entries for algebraic numbers, degree 4.

Formula

Equals (sqrt(7+2*sqrt(2))-sqrt(2)-1)/2.

Equals 1/A188636.

Smallest positive real root of x^4 + 2*x^3 - 3*x^2 - 2*x + 1 = 0.

Example

0.360409337131394214396560913709779476095838...

Mathematica

RealDigits[(Sqrt[7+2*Sqrt[2]]-Sqrt[2]-1)/2, 10, 100][[1]] (* Stefano Spezia, Apr 01 2026 *)

Prog

(PARI) (sqrt(7+2*sqrt(2))-sqrt(2)-1)/2
(PARI) polrootsreal(Pol([1, 2, -3, -2, 1]))[3]
(PARI) solve(x = 0.01, 1, log(x + 1 + sqrt(2)) + log(x))

Crossrefs

Cf. A014176, A188636.

Similar constants defined by continued fractions: A010527 (real part for r=i), A086773 (r=Pi), A086774 (r=exp(1)), A086775 (r=(1+sqrt(5))/2), A101263 (r=sqrt(2)), A188485-1 (r=1/2), A340616 (r=2*sqrt(2)).

Sequence in context: A334477 A278488 A396235 * A367325 A038023 A008954

Adjacent sequences: A394762 A394763 A394764 * A394766 A394767 A394768

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 31 2026

Status

approved