OFFSET
0,1
COMMENTS
See A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.
A 2*Pi-extension rectangle matches the continued fraction [6,2,3,1,1,3,2,1,16,47,...] of the shape L/W = Pi + sqrt(1 + Pi^2). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for the (2*Pi)-extension rectangle, 6 squares are removed first, then 2 squares, then 3 squares, then 1 square, then 1 square, ..., so that the original rectangle is partitioned into an infinite collection of squares.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..9999
EXAMPLE
6.4385009630654083972232325635946917292621665408132...
MAPLE
with(numtheory): cfrac(Pi+sqrt(1+Pi^2), 120, 'quotients'); # Muniru A Asiru, Nov 22 2018
MATHEMATICA
r = 2*Pi; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188725 *)
ContinuedFraction[t, 120] (* A188726 *)
ContinuedFraction[Pi + Sqrt[1 + Pi^2], 100] (* G. C. Greubel, Oct 31 2018 *)
PROG
(PARI) default(realprecision, 100); contfrac(Pi + sqrt(1 + Pi^2)) \\ G. C. Greubel, Oct 31 2018
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R) + Sqrt(1 + Pi(R)^2)); // G. C. Greubel, Oct 31 2018
CROSSREFS
KEYWORD
nonn,cofr
AUTHOR
Clark Kimberling, Apr 09 2011
STATUS
approved