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A188593
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Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.
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7
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1, 9, 0, 2, 1, 1, 3, 0, 3, 2, 5, 9, 0, 3, 0, 7, 1, 4, 4, 2, 3, 2, 8, 7, 8, 6, 6, 6, 7, 5, 8, 7, 6, 4, 2, 8, 6, 8, 1, 1, 3, 9, 7, 2, 6, 8, 2, 5, 1, 5, 0, 0, 4, 4, 4, 8, 9, 4, 6, 1, 1, 2, 8, 8, 8, 6, 0, 3, 0, 6, 3, 4, 0, 1, 7, 0, 3, 8, 7, 0, 0, 3, 4, 3, 7, 5, 8, 5, 6, 2, 1, 9, 4, 1, 6, 2, 2, 7, 6, 3, 3, 5, 1, 7, 9, 9, 4, 3, 5, 1, 0, 2, 8, 0, 6, 0, 0, 8, 4, 1, 7, 9, 7, 4, 1, 3, 2, 3, 8, 7
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OFFSET
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1,2
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COMMENTS
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A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012
Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014
The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020
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LINKS
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Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein's World of Mathematics, Golden Rectangle
Eric Weisstein's World of Mathematics, Pentagram
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FORMULA
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Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020
Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021
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EXAMPLE
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1.902113032590307144232878666758764286811397268251...
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MATHEMATICA
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r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130]][[1]]
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PROG
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(PARI) sqrt((5+sqrt(5))/2)
(MAGMA) SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018
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CROSSREFS
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Cf. A001622 (decimal expansion of the golden ratio), A019881.
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
Cf. A195693, A195723.
Sequence in context: A221429 A221507 A089481 * A065421 A198556 A261169
Adjacent sequences: A188590 A188591 A188592 * A188594 A188595 A188596
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KEYWORD
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nonn,cons,easy
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AUTHOR
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Clark Kimberling, Apr 04 2011
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STATUS
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approved
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