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*Pi^(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
KEYWORD
sign,frac
AUTHOR
Ozaner Hansha, Apr 16 2018
EXTENSIONS
More terms from Amiram Eldar, Jan 31 2019
STATUS
approved