A190157 Decimal expansion of (1+sqrt(-1+2*sqrt(5)))/2.

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

Offset

1,2

Comments

Let R denote a rectangle whose shape (i.e., length/width) is (1+sqrt(-1+2*sqrt(5)))/2. This rectangle can be partitioned into squares and golden rectangles in a manner that matches the periodic continued fraction [1,r,1,r,1,r,1,r,...], where r is the golden ratio. It can also be partitioned into squares so as to match the nonperiodic continued fraction [1,2,3,6,3,...] at A190158. For details, see A188635.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Example

1.431683416590579253079569133490735199410...

Mathematica

r = (1 + 5^(1/2))/2;
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190158 *)
RealDigits[N[%%, 120]]     (* A190157 *)
N[%%%, 40]
RealDigits[(1+Sqrt[-1+2*Sqrt[5]])/2, 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

Prog

(PARI) (1+sqrt(-1+2*sqrt(5)))/2 \\ G. C. Greubel, Dec 28 2017
(Magma) [(1+Sqrt(-1+2*Sqrt(5)))/2]; // G. C. Greubel, Dec 28 2017

Crossrefs

Cf. A188635, A190158, A189970, A189971.

Sequence in context: A253182 A283299 A355182 * A103552 A127673 A016698

Adjacent sequences: A190154 A190155 A190156 * A190158 A190159 A190160

Keyword

nonn,cons

Author

Clark Kimberling, May 05 2011

Status

approved