A091007 Decimal expansion of Sum_{n>=1} arccot(n^2).

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

Offset: 1

Author

Eric W. Weisstein, Dec 13 2003

Keywords

• nonn
• cons

Examples

			1.424741778429980889761547806889234152802066334601818040657245773...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• George Boros and Victor H. Moll, Sums of arctangents and some formulas of Ramanujan, Scientia, Ser. A: Math. Sci., Vol. 11 (2005) pp. 13-24, MR2196063, eq. (1.3).
• A. Sarkar, The sum of arctangents of reciprocal squares, Amer. Math. Monthly, 98, (1990), 652-653.
• Eric Weisstein's World of Mathematics, Inverse Cotangent.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Arctan((1-Cot(Pi(R)/Sqrt(2))*Tanh(Pi(R)/Sqrt(2)))/(1+Cot(Pi(R)/Sqrt(2))*Tanh(Pi(R)/Sqrt(2)))); // G. C. Greubel, Feb 01 2019

Mathematica

t = Cot[Pi Sqrt[2]/2] Tanh[Pi Sqrt[2]/2]; s = ArcCot[(1 + t)/(1 - t)]; RealDigits[N[s, 102]] (* Artur Jasinski, Sep 25 2008 *)

PARI

default(realprecision, 100); {t = cotan(Pi/sqrt(2))*tanh(Pi/sqrt(2))}; atan((1-t)/(1+t)) \\ G. C. Greubel, Feb 01 2019

Sage

t = cot(pi/sqrt(2))*tanh(pi/sqrt(2)); numerical_approx(atan((1-t)/(1+t)), digits=100) # G. C. Greubel, Feb 01 2019

Formula

Decimal expansion of transcendental number arccot((1 + t)/(1 - t)) where t=cot(Pi*sqrt(2)/2) tanh(Pi*sqrt(2)/2). - Artur Jasinski, Sep 25 2008

Equals Sum_{k>=1} (-1)^(k+1)*zeta(4*k-2)/(2*k-1). - Amiram Eldar, Mar 25 2021