A188720 Decimal expansion of (e+sqrt(4+e^2))/2.

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

Offset: 1

Author

Clark Kimberling, Apr 09 2011

Keywords

• nonn
• cons

Comments

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

An e-extension rectangle matches the continued fraction A188721 of the shape L/W = (1/2) *(e+sqrt(4+e^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-extension rectangle, 3 squares are removed first, then 21 squares, then 2 squares, then 40 squares, then 1 square,..., so that the original rectangle is partitioned into an infinite collection of squares.

(e+sqrt(4+e^2))/2 = [e,e,e,... ] (continued fraction). - Clark Kimberling, Sep 23 2013

Examples

			3.046524695333472471811401766587155243274607058879794774577422496312...
		

Links

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

Crossrefs

Cf. A188640, A188721, A188726, A188722, A188725.

Programs

Magma

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

Maple

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

Mathematica

r=E; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(E+Sqrt[4+E^2])/2,10,150][[1]] (* Harvey P. Dale, Jan 07 2015 *)

PARI

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