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 A257451 #30 Aug 27 2025 11:33:32 %S A257451 2,3,3,1,1,2,2,3,7,0,4,1,4,4,2,2,6,1,3,6,6,7,8,3,5,9,5,5,9,1,7,1,2,1, %T A257451 3,3,8,2,6,9,0,7,7,6,9,5,3,8,6,1,1,4,5,7,5,1,0,9,7,3,7,2,9,3,3,9,3,2, %U A257451 3,0,8,1,7,4,3,2,7,1,6,6,7,3,8,4,2,1,5,4,2,5,7,1,0,4,3,9,3,0,1,4,0,8,7,4,5 %N A257451 Decimal expansion of the location of the maximum of (1-cos(x))/x. %C A257451 Also, the first positive solution of x*sin(x)=(1-cos(x)). %C A257451 The function hsinc(x) = (1-cos(x))/x is the Hilbert transform of sinc(x) = sin(x)/x. Both functions play a considerable role in various branches of physics, particularly in spectroscopy. %C A257451 The value of hsinc(a) is in A257452. %C A257451 The kissing points [x,y] of the two tangents with the smallest nonzero |x|, drawn from the apex [0,1] of the function y = cos(x) to itself, have the coordinates [+a,cos(a)] and [-a,cos(a)], respectively. The angle each of the tangents subtends with the Y axis is theta = atan(1/sin(a)). - _Stanislav Sykora_, Oct 17 2015 %C A257451 For a curve S in the xy-plane starting at the origin, pointing to the right, turning counterclockwise with constant curvature K, and with an arclength of 1, let Y denote the maximum y-value of any point in S. Then, this constant is equal to the value of K that maximizes Y. - _Andrew Slattery_, Sep 11 2021 %C A257451 The smallest positive fixed point of tan(x/2). - _Michal Paulovic_, Aug 27 2025 %H A257451 Stanislav Sykora, <a href="/A257451/b257451.txt">Table of n, a(n) for n = 1..2000</a> %e A257451 2.3311223704144226136678359559171213382690776953861145751... %e A257451 Added in support of the Oct 17 2015 comment: %e A257451 cos(a) = -0.689157736645164443889295..., theta = atan(1/sin(a)) = 0.943742927149971739026594... rad = 54.072486671015691988683987... deg. %t A257451 RealDigits[x/.FindMaximum[(1-Cos[x])/x,x,WorkingPrecision->200] [[-1]]] [[1]] (* _Harvey P. Dale_, Mar 29 2022 *) %o A257451 (PARI) a = solve(x=1,3,x*sin(x)-1+cos(x)) %Y A257451 Cf. A257452. %K A257451 nonn,cons,changed %O A257451 1,1 %A A257451 _Stanislav Sykora_, Apr 23 2015