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 A254757 #20 Sep 01 2017 20:13:37
%S A254757 17,103,601,3503,20417,118999,693577,4042463,23561201,137324743,
%T A254757 800387257,4664998799,27189605537,158472634423,923646201001,
%U A254757 5383404571583,31376781228497,182877282799399,1065886915567897,6212444210607983
%N A254757 Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (-1, 5).
%C A254757 The corresponding y solutions are given in A220414.
%C A254757 The other part of the proper (sometimes called primitive) solutions are given in (A254758(n), A254759(n)) for n >= 1.
%C A254757 The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 0.
%D A254757 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%H A254757 Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/BinQuadForm.html">Binary Quadratic Forms (indefinite case)</a>.
%H A254757 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -1).
%H A254757 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A254757 a(n) = rational part of z(n), where z(n) = (-1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 1.
%F A254757 G.f.: (17 + x)/(1 - 6*x + x^2).
%F A254757 a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(0) = -1 and a(1) = 17.
%F A254757 a(n) = 17*S(n-1, 6) + S(n-2, 6), n >= 1, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).
%e A254757 The first pairs of positive solutions of this part of the Pell equation x^2 - 2*y^2 = - 7^2 are: [17, 13], [103, 73], [601, 425], [3503, 2477], [20417, 14437], [118999, 84145], [693577, 490433], [4042463, 2858453], [23561201, 16660285], [137324743, 97103257], ...
%t A254757 LinearRecurrence[{6,-1},{17,103},20] (* _Harvey P. Dale_, Sep 01 2017 *)
%o A254757 (PARI) Vec((17 + x)/(1 - 6*x + x^2) + O(x^30)) \\ _Michel Marcus_, Feb 08 2015
%Y A254757 Cf. A049310, A220414, A254758, A254759, A001653, A002315.
%K A254757 nonn,easy
%O A254757 1,1
%A A254757 _Wolfdieter Lang_, Feb 07 2015