A189964 - OEIS

A189964

Decimal expansion of (3+x+sqrt(38+6*x))/4, where x=sqrt(13).

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

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OFFSET

1,1

COMMENTS

This constant is the shape of a rectangle whose continued fraction partition matches [r,r,r,...], where r=(3+sqrt(13))/2. For a general discussion, see A188635. The ordinary continued fraction of r is [3,3,3,3,3,3,3,3,3,3,...]. A rectangle of shape r (that is, (length/width)=r) may be compared with the golden rectangle, with shape [1,1,1,1,1,1,...], and the silver rectangle, with shape [2,2,2,2,2,2,...].

LINKS

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

EXAMPLE

3.5819529507118503770725396959210446869118915483494611612922...

MATHEMATICA

r = (3 +13^(1/2))/2;

FromContinuedFraction[{r, {r}}]

FullSimplify[%]

N[%, 150]

RealDigits[%] (*A189964*)

ContinuedFraction[%%, 120] (*A189965*)

PROG