A188726 Continued fraction of the shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2).

6, 2, 3, 1, 1, 3, 2, 1, 16, 47, 1, 4, 2, 7, 1, 5, 317, 4, 1, 1, 1, 2, 13, 1, 38, 37, 1, 4, 1, 13, 1, 59, 3, 17, 1, 2, 2, 2, 5, 1, 3, 1, 3, 9, 1, 3, 4, 1, 2, 2, 1, 1, 2, 1, 23, 8, 9, 84, 1, 3, 1, 2, 1, 1, 3, 5, 5, 1, 1, 16, 1, 8, 4, 11, 1, 3, 1, 16, 4, 1, 1, 1, 1, 18, 1, 12, 1, 21, 3, 3, 1, 2, 4, 2, 10, 3, 5, 6, 1, 1, 25, 4, 10, 1, 5, 2, 1, 4, 16, 2, 5, 4, 2, 1, 4, 1, 1, 2, 1, 1

Offset: 0

Author

Clark Kimberling, Apr 09 2011

Keywords

• nonn
• cofr

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.

Examples

			6.4385009630654083972232325635946917292621665408132...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..9999

Crossrefs

Cf. A188640, A188725 (decimal expansion).

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R) + Sqrt(1 + Pi(R)^2)); // G. C. Greubel, Oct 31 2018

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 *)

PARI

default(realprecision, 100); contfrac(Pi + sqrt(1 + Pi^2)) \\ G. C. Greubel, Oct 31 2018