A190184 - OEIS

A190184

Decimal expansion of sqrt(1+x+sqrt(1+2*x)), where x=sqrt(2/3).

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

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OFFSET

1,2

COMMENTS

The rectangle R whose shape (i.e., length/width) is sqrt(1+x+sqrt(1+2x)), where x=sqrt(2/3), can be partitioned into rectangles of shapes sqrt(2) and sqrt(3) in a manner that matches the periodic continued fraction [sqrt(2), sqrt(3), sqrt(2), sqrt(3),...]. R can also be partitioned into squares so as to match the nonperiodic continued fraction [1,1,5,1,6,1,5,1,1,...] at A190185. For details, see A188635.

LINKS

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

EXAMPLE

1.854493630042558263683640132452778477778...

MATHEMATICA

FromContinuedFraction[{2^(1/2), 3^(1/2), {2^(1/2), 3^(1/2)}}]

FullSimplify[%]

ContinuedFraction[%, 100] (* A190185 *)

RealDigits[N[%%, 120]] (* A190186 *)

N[%%%, 40]

RealDigits[Sqrt[1+Sqrt[2/3]+Sqrt[1+2*Sqrt[2/3]]], 10, 100][[1]] (* G. C. Greubel, Dec 28 2017 *)

PROG