

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
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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



