This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A302977 #43 Jun 18 2025 16:54:12
%S A302977 1,-1,-1,-43,-223,-60623,-764783,-107351407,-2499928867,-596767688063,
%T A302977 -22200786516383,-64470807442488761,-3504534741776035061,
%U A302977 -3597207408242668198973,-268918457620309807441853,-185388032403184965693274807,-18241991360742724891839902347
%N A302977 Numerators of the rational factor of Kaplan's series for the Dottie number.
%C A302977 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.
%H A302977 Amiram Eldar, <a href="/A302977/b302977.txt">Table of n, a(n) for n = 0..100</a>
%H A302977 J. Bertrand, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k6570951r/f297.item.texteImage">Exercise III</a>, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
%H A302977 Ozaner Hansha, <a href="https://ozaner.github.io/dottie-number">The Dottie Number</a>.
%H A302977 Ozaner Hansha, <a href="https://ozaner.github.io/dottie-number/#kaplans-series">Kaplan's series</a>
%H A302977 Samuel R. Kaplan, <a href="https://web.archive.org/web/20201112024420/https://www.maa.org/sites/default/files/Kaplan2007-131105.pdf">The Dottie Number</a>, Math. Magazine, 80 (No. 1, 2007), 73-74.
%H A302977 V. Salov, <a href="https://arxiv.org/abs/1212.1027">Inevitable Dottie Number. Iterals of cosine and sine</a>, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
%F A302977 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).
%F A302977 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.
%e A302977 The partial Kaplan series at n=3 is d = Pi/4 - Pi^3/768 - Pi^5/61440 - 43*Pi^7/165150720.
%t A302977 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 *)
%Y A302977 Cf. A003957, A177413, A182503, A200309, A212112, A212113.
%Y A302977 Cf. A306254 (denominators).
%K A302977 sign,frac
%O A302977 0,4
%A A302977 _Ozaner Hansha_, Apr 16 2018
%E A302977 More terms from _Amiram Eldar_, Jan 31 2019