

A188738


Decimal expansion of esqrt(e^21).


20



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

0,2


COMMENTS

Decimal expansion of the shape of a lesser 2econtraction rectangle.
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
Suppose that ABCD is given and that the shape r=[ABCD] exceeds 2. The "rcontraction rectangles" of ABCD are here introduced as the rectangles AEFD and EBCF for which [AEFD]=[EBCF] and AE<>EB. That is, ABCD has the prescribed shape r, and AEFD and EBCF are mutually similar without being congruent. It is easy to prove that [AEFD]=(1/2)(rsqrt(4+r^2)) or [AEFD]=(1/2)(r+sqrt(4+r^2)); in the former case, we call AEFD the "lesser rcontraction rectangle", and the latter, the "greater rcontraction rectangle".
Both rcontraction rectangles match the continued fraction of [AEFD] in the following way. Write the continued fraction as [a(1),a(2),a(3),...]. Then, in the manner in which the continued fraction [1,1,1,...] matches the stepbystep removal of single squares from a golden triangle (as well as the manner in which the continued fraction [2,2,2,...] matches the stepbystep removal of 2 squares at a time from a silver triangle, etc.), remove a(1) squares at step 1, then remove a(2) squares at step 2, and so on, obtaining in the limit a partition of AEFD as an infinite set of squares.
For (related) rextension rectangles, see A188640.


LINKS

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


EXAMPLE

0.190623604147330614259428256541555268663022202.. = 1/A188739 with continued fraction 0, 5, 4, 15, 6, 1, 13, 2, 1, 1, 21, 3, 2, 16, 1, 4, 1, 1, 157,...


MAPLE

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


MATHEMATICA

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


PROG

(PARI) default(realprecision, 100); exp(1)  sqrt(exp(2)1) \\ G. C. Greubel, Nov 01 2018
(MAGMA) SetDefaultRealField(RealField(100)); Exp(1)  Sqrt(Exp(2)1); // G. C. Greubel, Nov 01 2018


CROSSREFS

Cf. A001113, A188739 (inverse), A188627 (continued fraction), A188640.
Sequence in context: A010533 A173201 A019740 * A199789 A019874 A197520
Adjacent sequences: A188735 A188736 A188737 * A188739 A188740 A188741


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Apr 11 2011


STATUS

approved



