The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A302977 Numerators of the rational factor of Kaplan's series for the Dottie number. 2
 1, -1, -1, -43, -223, -60623, -764783, -107351407, -2499928867, -596767688063, -22200786516383, -64470807442488761, -3504534741776035061, -3597207408242668198973, -268918457620309807441853, -185388032403184965693274807, -18241991360742724891839902347 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS In Kaplan's original article, where the term "Dottie" was coined, he mentioned that while the number was indeed transcendental, it was possible to express it as an infinite sum with the general term r_n*π^(2n+1) where r_n was a sequence of rational numbers. REFERENCES Bertrand, J., Exercise III in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie de L. Hachette et Cie, p. 285, 1865. LINKS Amiram Eldar, Table of n, a(n) for n = 0..100 Ozaner Hansha, The Dottie Number. Ozaner Hansha, Kaplan's series Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74. V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012. FORMULA These are the numerators of the unique sequence of rational numbers r_n such that d=Sum_{n>=0} (r_n*Pi^(2n+1)) (where d is the Dottie number A003957). r_0 = 1/4 and for n>0, r_n = b_(2n+1); where b_n = g^(n)(Pi/2)/(2^n*n!)) (and g^(n) is the n-th derivative of the inverse of x - cos x. A proof of this can be found in the second Hansha link. EXAMPLE The partial Kaplan series at n=3 is d = Pi/4 - Pi^3/768 - Pi^5/61440 - 43*Pi^7/165150720. MATHEMATICA f[x_] := x - Cos[x]; g[x_] := InverseFunction[f][x]; s = {Pi/4}; Do[AppendTo[s, Numerator[(-1/2)^n * 1/n! * Derivative[n][g][Pi/2]], {n, 3, 30, 2}]; s (* Amiram Eldar, Jan 31 2019 *) CROSSREFS Cf. A003957, A177413, A182503, A200309, A212112, A212113. Sequence in context: A038479 A142334 A107697 * A158080 A142450 A259421 Adjacent sequences:  A302974 A302975 A302976 * A302978 A302979 A302980 KEYWORD sign,frac AUTHOR Ozaner Hansha, Apr 16 2018 EXTENSIONS More terms from Amiram Eldar, Jan 31 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 16 05:18 EDT 2022. Contains 353693 sequences. (Running on oeis4.)