A231863 -id:A231863 - cp's OEIS frontend

A382242 Decimal expansion of Gamma(1/4)^2/(8sqrt(2Pi)).

6, 5, 5, 5, 1, 4, 3, 8, 8, 5, 7, 3, 0, 2, 9, 9, 5, 2, 6, 1, 6, 2, 0, 9, 8, 9, 7, 4, 7, 2, 7, 7, 9, 8, 5, 3, 4, 2, 0, 6, 8, 8, 7, 3, 7, 8, 5, 7, 9, 0, 5, 7, 9, 0, 7, 0, 4, 2, 0, 5, 4, 2, 5, 9, 5, 0, 1, 9, 7, 6, 4, 6, 7, 6, 7, 6, 0, 3, 5, 6, 2, 5, 5, 7, 5, 7, 3, 8, 8, 3, 2, 4, 0, 3, 5, 7, 2, 7, 3, 3, 6, 1, 5, 3, 3, 9, 3, 8, 1, 6, 7, 9, 4, 5, 8

Offset: 0

R. J. Mathar, Mar 19 2025

			0.6555143885730299526162098974727798534...

Leyda Almodovar, Victor H. Moll, Hadrian Quan, Fernando Roman, Eric Rowland, and Michole Washington, Infinite products arising in paperfolding, JIS 19 (2016), Article 16.5.1, eq. (72).

Cf. A034937, A068466, A231863, A005843, A005408.

Digits := 120 ; GAMMA(1/4)^2/8/sqrt(2*Pi) ; evalf(%) ;

RealDigits[Gamma[1/4]^2/(8*Sqrt[2*Pi]), 10, 120][[1]] (* Amiram Eldar, Mar 20 2025 *)

Equals A068466^2 *A231863 /8.

Equals Product_{n>=1} (A005843(n)/A005408(n))^A034947(n).

A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

Original entry on oeis.org

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, 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, 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

Offset: 0

Amiram Eldar, Aug 22 2025

			0.49169967769382111771654625416890810022151027126875...

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 61st William Lowell Putnam Mathematical Competition Saturday, December 2, 2000, The Putnam Archive. See Problem A-4.
Editors of the Mathematics Magazine, The 61st Annual William Lowell Putnam Examination, News and Letters, Mathematics Magazine, Vol. 74, No. 1 (2001), pp. 75-83. See Problem A4, pages 77 and 79-80.
Jihyung Kang and others, Convergence of I = Integral_0^oo sinx sin(x2) dx, Mathematics StackExchange, 2017.
Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson, The Sixty-first William Lowell Putnam Mathematical Competition, The American Mathematical Monthly, Vol. 108, No. 9 (2001), pp. 841-850. See pages 843 and 846, Problem A4.
Eric Weisstein's World of Mathematics, Fresnel Integrals.
Wikipedia, Fresnel integral.

Cf. A069998, A217481, A231863.

RealDigits[Integrate[Sin[x]*Sin[x^2], {x, 0, Infinity}], 10, 120][[1]]
(* or *)
RealDigits[Sqrt[Pi/2] * (Cos[1/4] * FresnelC[1/Sqrt[2*Pi]] + Sin[1/4] * FresnelS[1/Sqrt[2*Pi]]), 10, 120][[1]]

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.

Equals Integral_{x=0..1/2} cos(x^2 - 1/4) dx.

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A382242 Decimal expansion of Gamma(1/4)^2/(8sqrt(2Pi)).

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A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

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cp's OEIS Frontend

A382242 Decimal expansion of Gamma(1/4)^2/(8*sqrt(2*Pi)).

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Author

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Maple

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A387213 Decimal expansion of Integral_{x>=0} sin(x) * sin(x^2) dx.

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Mathematica

Formula

A382242 Decimal expansion of Gamma(1/4)^2/(8sqrt(2Pi)).