OFFSET
1,2
COMMENTS
Decimal expansion of the length/width ratio of a (2/Pi)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.
A (2/Pi)-extension rectangle matches the continued fraction [1,2,1,2,1,1,3,1,1,5,1,7,1,1,23,2,...] for the shape L/W = (1 + sqrt(1 + Pi^2))/Pi. 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 (2/Pi)-extension rectangle, 1 square is removed first, then 2 squares, then 1 square, then 2 squares, ..., so that the original rectangle of shape (1 + sqrt(1 + Pi^2))/Pi is partitioned into an infinite collection of squares.
EXAMPLE
1.36774839493136744469969176568220545565111326890...
MATHEMATICA
r = 2/Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
PROG
(PARI) (sqrt(Pi^2+1)+1)/Pi \\ Charles R Greathouse IV, Oct 01 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Apr 12 2011
STATUS
approved