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A188726 Continued fraction of the shape of a (2*Pi)-extension rectangle; shape = Pi + sqrt(1 + Pi^2). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 = 1..10000

FORMULA

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

Cf. A188640, A188727.

Sequence in context: A010133 A065280 A247672 * A272354 A196552 A062614

Adjacent sequences:  A188723 A188724 A188725 * A188727 A188728 A188729

KEYWORD

nonn,cofr,changed

AUTHOR

Clark Kimberling, Apr 09 2011

STATUS

approved

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Last modified April 23 06:03 EDT 2021. Contains 343199 sequences. (Running on oeis4.)