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A195693
Decimal expansion of arctan(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).
7
5, 5, 3, 5, 7, 4, 3, 5, 8, 8, 9, 7, 0, 4, 5, 2, 5, 1, 5, 0, 8, 5, 3, 2, 7, 3, 0, 0, 8, 9, 2, 6, 8, 5, 2, 0, 0, 3, 5, 0, 2, 3, 8, 2, 2, 7, 0, 0, 7, 1, 6, 3, 2, 3, 3, 3, 8, 2, 6, 9, 6, 0, 3, 7, 1, 6, 8, 5, 5, 1, 6, 9, 4, 8, 8, 6, 8, 1, 3, 9, 7, 0, 0, 6, 7, 0, 8, 5, 6, 4, 3, 4, 3, 0, 8, 5, 3, 2, 0, 7
OFFSET
0,1
COMMENTS
Radian measure of half the smaller angle in the golden rhombus. - Eric W. Weisstein, Dec 11 2018
The angle between the diagonal and the longer side of a golden rectangle. - Amiram Eldar, May 18 2021
LINKS
Paul S. Bruckman, Problem H-549, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 91; Resurrection, Solution to Problem H-549 by Charles K. Cook, ibid., Vol. 38, No. 2 (2000), pp. 191-192.
Hei-Chi Chan, Machin-type formulas expressing Pi in terms of phi, The Fibonacci Quarterly, Vol. 46/47, No. 1 (2008/2009), pp. 32-37.
Verner E. Hoggatt, Jr. and I. D. Bruggles, A Primer on the Fibonacci Sequence, Part V, The Fibonacci Quarterly, Vol. 2, No. 1 (1964), pp. 59-65.
Eric Weisstein's World of Mathematics, Golden Rhombus.
FORMULA
Equals Pi/2 - A195723. - Amiram Eldar, May 18 2021
Equals arctan(2)/2. - Christoph B. Kassir, Dec 04 2021
From Amiram Eldar, Jan 11 2022: (Start)
Equals arccot(phi).
Equals (Pi - arctan(phi^5))/3.
Equals (Pi - arctan(4/3))/4.
Equals Sum_{k>=1} ((-1)^(k+1) * arctan(1/Fibonacci(2*k))) (Bruckman, 1999). (End)
Equals Sum_{k>=1} arctan(1/Lucas(2*k)) (Hoggatt and Bruggles, 1964). - Amiram Eldar, Feb 05 2022
EXAMPLE
arctan(1/phi) = 0.5535743588970452515085327300892685200... .
tan(0.5535743588970452515085327300...) = 1/(golden ratio).
cot(0.5535743588970452515085327300...) = (golden ratio).
MATHEMATICA
(See also A195692.)
RealDigits[ArcCot[GoldenRatio], 10, 100][[1]] (* or *) RealDigits[(Pi - ArcTan[4/3])/4, 10, 100][[1]] (* Eric W. Weisstein, Dec 11 2018 *)
PROG
(PARI) atan(2)/2 \\ Michel Marcus, Feb 05 2022
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Sep 22 2011
STATUS
approved