A189970 - OEIS

A189970

Decimal expansion of (1 + x + sqrt(14+10*x))/4, where x=sqrt(5).

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

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OFFSET

1,1

COMMENTS

Let R denote a rectangle whose shape (i.e., length/width) is (1 + x + sqrt(14+10*x))/4, where x=sqrt(5). This rectangle can be partitioned into golden rectangles and squares in a manner that matches the periodic continued fraction [r,1,r,1,r,1,r,1,...]. It can also be partitioned into squares so as to match the nonperiodic continued fraction [2,3,6,3,...] at A189971. For details, see A188635.

Decimal expansion of sqrt(r + r*sqrt(r + r*sqrt(r + ...))), where r = (1 + sqrt(5))/2 = golden ratio. - Ilya Gutkovskiy, Aug 24 2015

A quartic integer. - Charles R Greathouse IV, Aug 29 2015

LINKS

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

Index entries for algebraic numbers, degree 4

EXAMPLE

2.31651242917313233045161321161782337624579...

MATHEMATICA

r = (1 + 5^(1/2))/2;

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

FullSimplify[%]

ContinuedFraction[%, 100] (* A189971 *)