

A190179


Decimal expansion of (1+sqrt(3+4*sqrt(2)))/2.


5



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

1,2


COMMENTS

Let R denote a rectangle whose shape (i.e., length/width) is (1+sqrt(3+4*sqrt(2)))/2. R can be partitioned into squares and silver rectangles in a manner that matches the periodic continued fraction [1,r,1,r,...], where r is the silver ratio: 1+sqrt(2)=[2,2,2,2,2,...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,3,5,1,2,1,1,1,2,...] at A190180. For details, see A188635.
The real value a1 is the only invariant point of the complexplane mapping M(c,z)=sqrt(csqrt(c+z)), with c = sqrt(2), and its only attractor, convergent from any starting complexplane location.  Stanislav Sykora, Apr 29 2016


LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10001


FORMULA

Equals 1+sqrt(csqrt(c+sqrt(csqrt(c+ ...)))), with c=sqrt(2).  Stanislav Sykora, Apr 29 2016


EXAMPLE

1.314992983020771197119164203638263044565...


MATHEMATICA

r = 1 + 2^(1/2));
FromContinuedFraction[{1, r, {1, r}}]
FullSimplify[%]
ContinuedFraction[%, 100] (* A190180 *)
RealDigits[N[%%, 120]] (* A190179 *)
N[%%%, 40]
RealDigits[(1+Sqrt[4Sqrt[2]3])/2, 10, 120][[1]] Harvey P. Dale, May 19 2012


PROG

(PARI) (1+sqrt(3+4*sqrt(2)))/2 \\ Altug Alkan, Apr 29 2016
(MAGMA) (1+Sqrt(3+4*Sqrt(2)))/2; // G. C. Greubel, Dec 28 2017


CROSSREFS

Cf. A188635, A190180, A190177, A190178.
Sequence in context: A202353 A108621 A193792 * A025116 A178300 A081720
Adjacent sequences: A190176 A190177 A190178 * A190180 A190181 A190182


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, May 05 2011


STATUS

approved



