A190262 - OEIS

A190262

Decimal expansion of (3 + sqrt(9 + 12x))/6, where x=sqrt(3).

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

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OFFSET

1,2

COMMENTS

The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+12x))/6, where x=sqrt(3), can be partitioned into rectangles of shapes 1 and sqrt(3) in a manner that matches the periodic continued fraction [1, x, 1, x, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1, 2, 2, 3, 1, 3, 2, 1, 1, 1, ...] at A190263. For details, see A188635.

LINKS

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

EXAMPLE

1.409587966713294731518226466119659876240...

MATHEMATICA

r=3^(1/2)

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

FullSimplify[%]

ContinuedFraction[%, 100] (* A190263 *)

RealDigits[N[%%, 120]] (* A190262 *)

N[%%%, 40]

RealDigits[(3 + Sqrt[9 + 12*Sqrt[3]])/6, 10, 100] (* G. C. Greubel, Dec 28 2017 *)