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 A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle. 9
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10001 Michael Penn, On the fifth root of the identity matrix., YouTube video, 2022. Eric Weisstein's World of Mathematics, Golden Rectangle Eric Weisstein's World of Mathematics, Pentagram Index entries for algebraic numbers, degree 4 FORMULA Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020 Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021 Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022 Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022 Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024 EXAMPLE 1.902113032590307144232878666758764286811397268251... MATHEMATICA r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130]][[1]] RealDigits[Sqrt[GoldenRatio+2], 10, 130][[1]] (* Harvey P. Dale, Oct 27 2023 *) PROG (PARI) sqrt((5+sqrt(5))/2) (Magma) SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018 CROSSREFS Cf. A001622 (decimal expansion of the golden ratio), A019881. Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)). Cf. A195693, A195723, A296184. Sequence in context: A221507 A370347 A089481 * A065421 A198556 A261169 Adjacent sequences: A188590 A188591 A188592 * A188594 A188595 A188596 KEYWORD nonn,cons,easy,changed AUTHOR Clark Kimberling, Apr 04 2011 STATUS approved

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Last modified February 29 15:18 EST 2024. Contains 370425 sequences. (Running on oeis4.)