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 A132338 #107 Jan 05 2025 19:51:38
%S A132338 3,8,1,9,6,6,0,1,1,2,5,0,1,0,5,1,5,1,7,9,5,4,1,3,1,6,5,6,3,4,3,6,1,8,
%T A132338 8,2,2,7,9,6,9,0,8,2,0,1,9,4,2,3,7,1,3,7,8,6,4,5,5,1,3,7,7,2,9,4,7,3,
%U A132338 9,5,3,7,1,8,1,0,9,7,5,5,0,2,9,2,7,9,2,7,9,5,8,1,0,6,0,8,8,6,2,5,1,5,2,4
%N A132338 Decimal expansion of 1 - 1/phi.
%C A132338 Density of 1's in Fibonacci word A003849.
%C A132338 Also decimal expansion of Sum_{n>=1} ((-1)^(n+1))*1/phi^n. - _Michel Lagneau_, Dec 04 2011
%C A132338 The Lambert series evaluated at this point is 0.8828541617125076... [see André-Jeannin]. - _R. J. Mathar_, Oct 28 2012
%C A132338 Because this equals 2 - phi, this is an integer in the quadratic number field Q(sqrt(5)). (Note that this is also sqrt(5 - 3*phi).) - _Wolfdieter Lang_, Jan 08 2018
%C A132338 When m >= 1, the equation m*x^m + (m-1)*x^(m-1) + ... + 2*x^2 + x - 1 = 0 has only one positive root, u(m) (say); then lim_{m->oo} u(m) = (3-sqrt(5))/2 (see Aubonnet). - _Bernard Schott_, May 12 2019
%C A132338 Cosine of the zenith angle at which a string should be cut so that a ball tied to one of its ends, set moving without friction around a vertical circle with the minimum speed in a uniform gravitational field, will then travel through the fixed center of the circle. - _Stefano Spezia_, Oct 25 2020
%C A132338 Algebraic number of degree 2 with minimal polynomial x^2 - 3*x + 1. The other root is 1 + phi = A104457. - _Wolfdieter Lang_, Aug 29 2022
%D A132338 F. Aubonnet, D. Guinin and A. Ravelli, Oral, Concours d'entrée des Grandes Ecoles Scientifiques, Exercices résolus, "Crus" 1982-83, Bréal, 1983, Exercice 210, 40-42.
%H A132338 Ivan Panchenko, <a href="/A132338/b132338.txt">Table of n, a(n) for n = 0..1000</a>
%H A132338 R. André-Jeannin, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/28-3/andre-jeannin.pdf">Lambert series and the summation of reciprocals in certain Fibonacci-Lucas-Type sequences</a>, Fib. Quart. 28 (1990) 223-226.
%H A132338 Yiyan Ni, Myron Hlynka, and Percy H. Brill, <a href="https://arxiv.org/abs/1806.09150">Urn Models and Fibonacci Series</a>, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.
%H A132338 <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F A132338 Equals 1 - 1/phi = 2 - phi, with phi from A001622.
%F A132338 Equals A094874 - 1, or A079585 - 2, or the square of A094214.
%F A132338 Equals (5-sqrt(5))^2/20 = 1/phi^2 = 1/A104457. - _Joost Gielen_, Sep 28 2013 [corrected by _Joerg Arndt_, Sep 29 2013]
%F A132338 Equals (3-sqrt(5))/2. - _Bernard Schott_, May 12 2019
%F A132338 Equals Sum_{k >= 2} (-1)^k/(Fibonacci(k)*Fibonacci(k+1)). See Ni et al. - _Michel Marcus_, Jun 26 2018
%e A132338 0.38196601125010515179541316563436188...
%t A132338 RealDigits[N[1/GoldenRatio^2,200]] (* _Vladimir Joseph Stephan Orlovsky_, May 27 2010 *)
%t A132338 RealDigits[1-1/GoldenRatio,10,120][[1]] (* _Harvey P. Dale_, Mar 30 2024 *)
%o A132338 (PARI) (3-sqrt(5))/2 \\ _Michel Marcus_, Oct 26 2020
%Y A132338 Cf. A001622, A003849, A094874, A079585, A094214, A104457.
%K A132338 cons,nonn
%O A132338 0,1
%A A132338 _N. J. A. Sloane_, Nov 07 2007