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 A118894 #7 Jan 05 2023 11:06:06 %S A118894 3,7,11,15,19,23,27,31,35,43,47,51,59,63,67,71,75,79,83,87,91,99,103, %T A118894 107,115,119,123,127,131,135,139,143,151,159,163,167,171,175,179,187, %U A118894 191,195,199,211,215,219,223,227,231,235,239,243,247,251,255,263,267 %N A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even. %C A118894 Numbers m such that A002350(m) is even. These m can be used to generate consecutive odd powerful numbers, as in A076445. As shown by Lang, the solution of Pell's equation is greatly simplified by Chebyshev polynomials of the first kind T(n,x), which is illustrated in A001075 for the case m=3. In that case, the solutions are x=T(n,2), for integer n>0. For any m in this sequence, let E(k)=T(m+2mk,A002350(m)). Then E(k)-1 and E(k)+1 are consecutive odd powerful numbers for k=0,1,2,... %H A118894 Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/p36pub/p36.pdf">Chebyshev Polynomials and Certain Quadratic Diophantine Equations</a> %H A118894 H. W. Lenstra Jr., <a href="http://www.ams.org/notices/200202/fea-lenstra.pdf">Solving the Pell equation</a>, Notices AMS, 49 (2002), 182-192. %Y A118894 Cf. A001075, A001091, A023038, A001081, A001085, A077424, A097310 (x solutions for m=3, 15, 35, 63, 99, 143, 195). %K A118894 nonn %O A118894 1,1 %A A118894 _T. D. Noe_, May 04 2006