

A188640


Decimal expansion of e + sqrt(1+e^2).


61



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

1,1


COMMENTS

The shape of a rectangle WXYZ, denoted by [WXYZ], is defined by length/width: [WXYZ]=max{WX/YZ, YZ/WX}. Consider the following configuration of rectangles AEFD, EBCF, ABCD, where AEFD is not a square:
D................F....C
.......................
.......................
.......................
A................E....B
Let r=[AEFD]. The rextension rectangle of AEFD is here introduced as the rectangle ABCD for which [AEFD]=[EBCF] and AE<>EB. That is, AEFD has the prescribed shape r, and AEFD and EBCF are similar without being congruent.
We extend the definition of rextension rectangle to the case that 0<r<1; in this case, [AEFD]=1/r and ABCD is defined as above.
Then for all r>0, it is easy to prove that [ABCD] = (r+sqrt(4+r^2))/2.
This here is the length/width ratio for the (2e)extension rectangle.
A (2e)extension rectangle matches the continued fraction A188796 for the shape L/W=(e+sqrt(1+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 the (2e)extension rectangle, 5 squares are removed first, then 1 square, then 1 square, then 1 square, then 1 square, then 2 squares..., so that the original rectangle is partitioned into an infinite collection of squares.
Shapes of other rextension rectangles, partitionable into a collection of squares in accord with the continued fraction of the shape [ABCD], are approximated at A188635A188639, A188655A188659, and A188720A188737.
For (related) rcontraction rectangles, see A188738 and A188739.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108109.
Clark Kimberling, Two kinds of golden triangles, generalized to match continued fractions, Journal for Geometry and Graphics, 11 (2007) 165171.
Index entries for transcendental numbers


EXAMPLE

Length/width = 5.61466856004905343925478283318633736023982...


MAPLE

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


MATHEMATICA

r=2E; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]


PROG

(PARI) exp(1)+sqrt(1+exp(2)) \\ Charles R Greathouse IV, Jun 16 2011
(MAGMA) SetDefaultRealField(RealField(100)); Exp(1) + Sqrt(1 + Exp(2)); // G. C. Greubel, Oct 31 2018


CROSSREFS

Cf. A188796, A188738, A188739, A188720.
Sequence in context: A298171 A046614 A080130 * A198419 A323738 A222133
Adjacent sequences: A188637 A188638 A188639 * A188641 A188642 A188643


KEYWORD

nonn,cons,easy


AUTHOR

Clark Kimberling, Apr 10 2011


STATUS

approved



