This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A195693 #51 Feb 16 2025 08:33:15
%S A195693 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,
%T A195693 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,
%U A195693 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
%N A195693 Decimal expansion of arctan(1/phi), where phi = (1+sqrt(5))/2 (the golden ratio).
%C A195693 Radian measure of half the smaller angle in the golden rhombus. - _Eric W. Weisstein_, Dec 11 2018
%C A195693 The angle between the diagonal and the longer side of a golden rectangle. - _Amiram Eldar_, May 18 2021
%H A195693 Paul S. Bruckman, <a href="https://fq.math.ca/Scanned/37-1/advanced37-1.pdf">Problem H-549</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 91; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-2/advanced38-2.pdf">Resurrection</a>, Solution to Problem H-549 by Charles K. Cook, ibid., Vol. 38, No. 2 (2000), pp. 191-192.
%H A195693 Hei-Chi Chan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/46_47-1/Chan_12-08.pdf">Machin-type formulas expressing Pi in terms of phi</a>, The Fibonacci Quarterly, Vol. 46/47, No. 1 (2008/2009), pp. 32-37.
%H A195693 Verner E. Hoggatt, Jr. and I. D. Bruggles, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/2-1/hoggatt2.pdf">A Primer on the Fibonacci Sequence, Part V</a>, The Fibonacci Quarterly, Vol. 2, No. 1 (1964), pp. 59-65.
%H A195693 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>.
%H A195693 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A195693 Equals Pi/2 - A195723. - _Amiram Eldar_, May 18 2021
%F A195693 Equals arctan(2)/2. - _Christoph B. Kassir_, Dec 04 2021
%F A195693 From _Amiram Eldar_, Jan 11 2022: (Start)
%F A195693 Equals arccot(phi).
%F A195693 Equals (Pi - arctan(phi^5))/3.
%F A195693 Equals (Pi - arctan(4/3))/4.
%F A195693 Equals Sum_{k>=1} ((-1)^(k+1) * arctan(1/Fibonacci(2*k))) (Bruckman, 1999). (End)
%F A195693 Equals Sum_{k>=1} arctan(1/Lucas(2*k)) (Hoggatt and Bruggles, 1964). - _Amiram Eldar_, Feb 05 2022
%e A195693 arctan(1/phi) = 0.5535743588970452515085327300892685200... .
%e A195693 tan(0.5535743588970452515085327300...) = 1/(golden ratio).
%e A195693 cot(0.5535743588970452515085327300...) = (golden ratio).
%t A195693 (See also A195692.)
%t A195693 RealDigits[ArcCot[GoldenRatio], 10, 100][[1]] (* or *) RealDigits[(Pi - ArcTan[4/3])/4, 10, 100][[1]] (* _Eric W. Weisstein_, Dec 11 2018 *)
%o A195693 (PARI) atan(2)/2 \\ _Michel Marcus_, Feb 05 2022
%Y A195693 Cf. A000032, A000045, A001622, A005248, A175288, A195692, A195694, A195723.
%K A195693 nonn,cons
%O A195693 0,1
%A A195693 _Clark Kimberling_, Sep 22 2011