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 A192222 #35 Jan 05 2025 19:51:39
%S A192222 1,2,5,34,1597,3524578,17167680177565,407305795904080553832073954,
%T A192222 229265413057075367692743352179590077832064383222590237
%N A192222 a(n) = Fibonacci(2^n + 1).
%C A192222 a(n) is the numerator of the n-th iterate when Newton's method is applied to the function x^2 - x - 1 with initial guess x = 1. The n-th iterate is a(n)/A058635(n). - _Jason Zimba_, Jan 20 2023
%H A192222 John Gill and Matthew Miller, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/19-1/gill.pdf">Newton's Method and Ratios of Fibonacci Numbers</a>, Fibonacci Quarterly, 19(1):1-3, February 1981.
%H A192222 Jonathan Sondow, <a href="https://doi.org/10.1063/1.3630044">Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers</a>, Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, Vol. 1385, No. 1 (2011), pp. 97-100, <a href="http://arxiv.org/abs/1106.4246">arXiv preprint</a>, arXiv:1106.4246 [math.NT], 2011.
%H A192222 Yohei Tachiya, <a href="http://dx.doi.org/10.1016/j.jnt.2006.11.006">Transcendence of certain infinite products</a>, J. Number Theory, Vol. 125, No. 1 (2007), pp. 182-200.
%F A192222 a(n) = A000045(2^n + 1).
%F A192222 Product_{n>0} (1 + 1/a(n)) = 3/phi = A134973, where phi = (1+sqrt(5))/2 is the golden mean.
%F A192222 Sum_{n>=0} 1/a(n) = A338305. - _Amiram Eldar_, Oct 22 2020
%t A192222 Table[Fibonacci[2^n + 1], {n, 0, 10}] (* _T. D. Noe_, Jan 11 2012 *)
%Y A192222 Cf. A000045 (Fibonacci numbers F(n)), A001622, A134973 (decimal expansion of 3/phi), A192223 (Lucas(2^n + 1)), A338305.
%K A192222 nonn,easy
%O A192222 0,2
%A A192222 _Jonathan Sondow_, Jun 26 2011