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 A387213 #8 Aug 23 2025 09:24:57
%S A387213 4,9,1,6,9,9,6,7,7,6,9,3,8,2,1,1,1,7,7,1,6,5,4,6,2,5,4,1,6,8,9,0,8,1,
%T A387213 0,0,2,2,1,5,1,0,2,7,1,2,6,8,7,5,5,0,7,7,2,5,5,9,0,4,8,1,7,9,1,4,7,4,
%U A387213 5,0,7,2,2,3,7,5,6,2,9,6,3,8,1,0,1,9,1,1,8,9,9,8,7,5,7,6,4,6,6,2,9,0,2,1,1
%N A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.
%H A387213 Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, <a href="https://kskedlaya.org/putnam-archive/2000s.pdf">Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000</a>, The Putnam Archive. See Problem A-4.
%H A387213 Editors of the Mathematics Magazine, <a href="https://www.jstor.org/stable/2691163">The 61st Annual William Lowell Putnam Examination</a>, News and Letters, Mathematics Magazine, Vol. 74, No. 1 (2001), pp. 75-83. See Problem A4, pages 77 and 79-80.
%H A387213 Jihyung Kang and others, <a href="https://math.stackexchange.com/questions/2281381/convergence-of-i-int-0-infty-sin-x-sinx2-mathrmdx">Convergence of I = Integral_0^oo sinx sin(x2) dx</a>, Mathematics StackExchange, 2017.
%H A387213 Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, <a href="https://www.jstor.org/stable/2695556">The Sixty-first William Lowell Putnam Mathematical Competition</a>, The American Mathematical Monthly, Vol. 108, No. 9 (2001), pp. 841-850. See pages 843 and 846, Problem A4.
%H A387213 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FresnelIntegrals.html">Fresnel Integrals</a>.
%H A387213 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fresnel_integral">Fresnel integral</a>.
%F A387213 Equals sqrt(Pi/2) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.
%F A387213 Equals Integral_{x=0..1/2} cos(x^2 - 1/4) dx.
%e A387213 0.49169967769382111771654625416890810022151027126875...
%t A387213 RealDigits[Integrate[Sin[x]*Sin[x^2], {x, 0, Infinity}], 10, 120][[1]]
%t A387213 (* or *)
%t A387213 RealDigits[Sqrt[Pi/2] * (Cos[1/4] * FresnelC[1/Sqrt[2*Pi]] + Sin[1/4] * FresnelS[1/Sqrt[2*Pi]]), 10, 120][[1]]
%Y A387213 Cf. A069998, A217481, A231863.
%K A387213 nonn,cons,new
%O A387213 0,1
%A A387213 _Amiram Eldar_, Aug 22 2025