A086775 - OEIS

A086775

Decimal expansion of the number defined by the continued fraction 1/(phi+1/(phi+1/(phi + ...))).

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

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OFFSET

0,1

COMMENTS

This number is the inverse of the number whose decimal expansion is A136319. It is also the latter number minus phi. - Geoffrey Caveney, Apr 18 2014

From A.H.M. Smeets, Apr 01 2026: (Start)

Repeat s = s + phi; s=1/s. The initial value of s is irrelevant, as long as s != -phi.

Similar constants defined by continued fractions 1/(r+1/(r+1/(r + ...))): A010527 (real part for r=i), A086773 (r=Pi), A086774 (r=exp(1)), A101263 (r=sqrt(2)), A188485-1 (r=1/2), A340616 (r=2*sqrt(2)), A394765 (r=(sqrt(7+2*sqrt(2))-sqrt(2)-1)/2).

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. (End)

LINKS

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

Index entries for algebraic numbers, degree 4.

FORMULA

Repeat s = s + phi; s=1/s. The initial value of s is irrelevant, as long as it isn't -phi.

Equals (sqrt(5+phi)-phi)/2, where phi = A001622. - R. J. Mathar, Sep 15 2012

Equals 1/A136319. - Geoffrey Caveney, Apr 18 2014

Smallest positive real root of x^4 + x^3 - 3*x^2 - x + 1 = 0. - A.H.M. Smeets, Mar 29 2026

EXAMPLE

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phi + 1

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