A302977 Numerators of the rational factor of Kaplan's series for the Dottie number.
1, -1, -1, -43, -223, -60623, -764783, -107351407, -2499928867, -596767688063, -22200786516383, -64470807442488761, -3504534741776035061, -3597207408242668198973, -268918457620309807441853, -185388032403184965693274807, -18241991360742724891839902347
Offset: 0
Examples
The partial Kaplan series at n=3 is d = Pi/4 - Pi^3/768 - Pi^5/61440 - 43*Pi^7/165150720.
Links
- Amiram Eldar, Table of n, a(n) for n = 0..100
- J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
- 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.
Programs
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Mathematica
f[x_] := x - Cos[x]; g[x_] := InverseFunction[f][x]; s = {1}; Do[AppendTo[s, Numerator[(-1/2)^n * 1/n! * Derivative[n][g][Pi/2]]], {n, 3, 30, 2}]; s (* Amiram Eldar, Jan 31 2019 *)
Formula
These are the numerators of the unique sequence of rational numbers r_n such that d = Sum_{n>=0} r_n*Pi^(2*n+1) (where d is the Dottie number A003957).
r_0 = 1/4 and for n>0, r_n = b_(2*n+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.
Extensions
More terms from Amiram Eldar, Jan 31 2019
Comments