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A190258 Decimal expansion of (x + sqrt(2 + 4x))/2, where x=sqrt(2). 3
2, 0, 9, 0, 6, 5, 7, 8, 5, 0, 8, 5, 2, 2, 4, 4, 7, 7, 5, 7, 1, 0, 0, 8, 9, 6, 3, 5, 0, 0, 5, 2, 2, 1, 3, 2, 8, 0, 9, 5, 8, 8, 0, 1, 7, 1, 5, 3, 5, 0, 8, 9, 6, 1, 5, 2, 7, 0, 1, 5, 4, 0, 8, 0, 1, 3, 6, 5, 3, 8, 6, 8, 6, 5, 8, 2, 3, 0, 1, 7, 6, 3, 7, 1, 1, 4, 3, 1, 5, 0, 4, 0, 4, 6, 0, 4, 2, 6, 3, 8, 4, 6, 7, 1, 8, 0, 8, 3, 2, 7, 8, 0, 6, 7, 6, 9, 3, 2, 5, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The rectangle R whose shape (i.e., length/width) is (x+sqrt(2+4x))/2, where x=sqrt(2), can be partitioned into rectangles of shapes sqrt(2) and 1 in a manner that matches the periodic continued fraction [x, 1, x, 1, ...].  R can also be partitioned into squares so as to match the nonperiodic continued fraction [2,11,32,1,4,10,2,1,...] at A190259.  For details, see A188635.

LINKS

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

EXAMPLE

2.090657850852244775710089635005221328095...

MATHEMATICA

r=2^(1/2);

FromContinuedFraction[{r, 1, {r, 1}}]

FullSimplify[%]

ContinuedFraction[%, 100]  (* A190258 *)

RealDigits[N[%%, 120]]     (* A190259 *)

N[%%%, 40]

PROG

(PARI) sqrt(1/2)+sqrt(1/2+sqrt(2))

(MAGMA) [(Sqrt(2) + Sqrt(2+4*Sqrt(2)))/2]; // G. C. Greubel, Dec 26 2017

CROSSREFS

Cf. A190259, A190260, A188635.

Sequence in context: A140415 A105819 A153616 * A161119 A019750 A308473

Adjacent sequences:  A190255 A190256 A190257 * A190259 A190260 A190261

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, May 06 2011

STATUS

approved

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Last modified November 14 22:45 EST 2019. Contains 329135 sequences. (Running on oeis4.)