

A188926


Decimal expansion of sqrt((7+sqrt(13))/6).


1



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

1,2


COMMENTS

Decimal expansion of the length/width ratio of a sqrt(1/3)extension rectangle. See A188640 for definitions of shape and rextension rectangle.
A sqrt(1/3)extension rectangle matches the continued fraction [1,3,28,1,2,2,42,1,1,1,4,...] for the shape L/W=sqrt((7+sqrt(13))/6). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(1/3)extension rectangle, 1 square is removed first, then 3 squares, then 28 squares, then 1 square,..., so that the original rectangle of shape sqrt((7+sqrt(13))/6) is partitioned into an infinite collection of squares.


LINKS



EXAMPLE

1.32950813432787924989572324374094447133596...


MATHEMATICA

r = 3^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[Sqrt[(7+Sqrt[13])/6], 10, 140][[1]] (* Harvey P. Dale, Feb 08 2013 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



