A188727 Decimal expansion of (e+sqrt(16+e^2))/4.

1, 8, 8, 8, 6, 2, 6, 2, 8, 9, 6, 4, 8, 2, 1, 6, 1, 6, 7, 0, 7, 5, 8, 1, 9, 4, 2, 5, 3, 2, 1, 7, 7, 0, 9, 2, 4, 4, 2, 4, 1, 9, 5, 2, 7, 0, 1, 1, 9, 0, 6, 0, 6, 0, 0, 9, 4, 2, 6, 4, 6, 6, 8, 8, 2, 5, 7, 9, 6, 8, 5, 5, 6, 1, 0, 1, 6, 9, 4, 5, 7, 4, 2, 8, 7, 0, 6, 2, 9, 9, 5, 7, 1, 6, 9, 2, 4, 5, 4, 1, 7, 5, 9, 0, 1, 3, 4, 9, 3, 3, 5, 7, 9, 1, 6, 1, 2, 2, 4, 6, 4, 3, 8, 9, 5, 4, 5, 0, 1, 8

Offset: 1

Author

Clark Kimberling, Apr 10 2011

Keywords

• nonn
• cons

Comments

Decimal expansion of shape of an (e/2)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, an r-extension rectangle is composed of two rectangles of shape r.

An (e/2)-extension rectangle matches the continued fraction A188728 of the shape L/W = (r+sqrt(4+r^2))/2. This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...]. Specifically, for an (e/2)-extension rectangle, 1 square is removed first, then 1 square, then 7 squares, then 1 square, then 46 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

Examples

			1.88862628964821616707581942532177092442419527...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A188640, A188727, A001113.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(1) + Sqrt(16 + Exp(2)))/4; // G. C. Greubel, Oct 31 2018

Maple

evalf((exp(1)+sqrt(16+exp(2)))/4,140); # Muniru A Asiru, Nov 01 2018

Mathematica

r = e/2; t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A188727 *)
ContinuedFraction[t, 120] (* A188728 *)

PARI

default(realprecision, 100); (exp(1) + sqrt(16 + exp(2)))/4 \\ G. C. Greubel, Oct 31 2018