A190285 - OEIS

A190285

Decimal expansion of (3+sqrt(9+4r))/2, where r=sqrt(3).

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

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OFFSET

1,1

COMMENTS

The rectangle R whose shape (i.e., length/width) is (3+sqrt(9+4r))/2, where r=sqrt(3), can be partitioned into rectangles of shapes 3 and r in a manner that matches the periodic continued fraction [3, r, 3, r, ...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [3,2,55,6,1,1,1,9,1,1,1,7,2,...] at A190286. For details, see A188635.

LINKS

Table of n, a(n) for n=1..120.

EXAMPLE

3.495507656604924503772866679054481005189...

MATHEMATICA

r=3^(1/2)

FromContinuedFraction[{3, r, {3, r}}]

FullSimplify[%]

ContinuedFraction[%, 100] (* A190286 *)

RealDigits[N[%%, 120]] (* A190285 *)

N[%%%, 40]

RealDigits[(3+Sqrt[9+4Sqrt[3]])/2, 10, 120][[1]] (* Harvey P. Dale, Oct 19 2021 *)