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 r-extension 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.
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
nonn,cons
AUTHOR
Clark Kimberling, Apr 13 2011
STATUS
approved