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A190157 Decimal expansion of (1+sqrt(-1+2*sqrt(5)))/2. 5
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 (list; constant; graph; refs; listen; history; text; internal format)
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: A087274 A253182 A283299 * A103552 A127673 A016698

Adjacent sequences:  A190154 A190155 A190156 * A190158 A190159 A190160

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, May 05 2011

STATUS

approved

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Last modified November 17 20:50 EST 2018. Contains 317278 sequences. (Running on oeis4.)