

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. 12, 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), 7374.
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 nth 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



