%I #57 Apr 08 2026 15:34:22

%S 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,

%T 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,

%U 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

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

%C 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

%C From _A.H.M. Smeets_, Apr 01 2026: (Start)

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

%C 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).

%C 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)

%H A.H.M. Smeets, <a href="/A086775/b086775.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.

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

%F Equals (sqrt(5+phi)-phi)/2, where phi = A001622. - _R. J. Mathar_, Sep 15 2012

%F Equals 1/A136319. - _Geoffrey Caveney_, Apr 18 2014

%F Smallest positive real root of x^4 + x^3 - 3*x^2 - x + 1 = 0. - _A.H.M. Smeets_, Mar 29 2026

%e 1

%e ------

%e phi + 1

%e ------

%e phi + 1

%e -------

%e phi + etc

%e Equals 0.4772599964740196445422298845...

%t First[RealDigits[(Sqrt[5 + GoldenRatio] - GoldenRatio)/2, 10, 100]] (* _Paolo Xausa_, Apr 07 2026 *)

%o (PARI) ?\p 2000 ?f(n) = phi=(sqrt(5)+1)/2; s=0; for(x=1,n,s=s+phi; s=1/s); print(s)

%o (PARI) polrootsreal(Pol([1,1,-3,-1,1]))[3] \\ _A.H.M. Smeets_, Mar 29 2026

%o (PARI) solve(x = 0.01, 1, log(x + (1 + sqrt(5))/2) + log(x)) \\ _A.H.M. Smeets_, Apr 01 2026

%Y Cf. A001622, A136319.

%K easy,nonn,cons,changed

%O 0,1

%A _Cino Hilliard_, Aug 03 2003

%E Name clarified by _Sean A. Irvine_, Mar 31 2026