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 A254758 #14 Jun 28 2019 07:15:36
%S A254758 1,23,137,799,4657,27143,158201,922063,5374177,31322999,182563817,
%T A254758 1064059903,6201795601,36146713703,210678486617,1227924205999,
%U A254758 7156866749377,41713276290263,243122790992201,1417023469662943
%N A254758 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 A254758 The corresponding y solutions are given in A254759.
%C A254758 The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.
%C A254758 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 >= 1.
%D A254758 T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
%H A254758 Wolfdieter Lang, <a href="http://www.itp.kit.edu/~wl/BinQuadForm.html">Binary Quadratic Forms (indefinite case)</a>.
%H A254758 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -1).
%H A254758 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A254758 a(n) = rational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.
%F A254758 G.f.: (1+17*x)/(1-6*x+x^2).
%F A254758 a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = -17 and a(0) = 1.
%F A254758 a(n) = S(n, 6) + 17*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).
%e A254758 The first pairs of positive solutions of this part of the Pell equation x^2 - 2*y^2 = - 7^2 are: [1, 5], [23, 17], [137, 97], [799, 565], [4657, 3293], [27143, 19193], [158201, 111865], [922063, 651997], [5374177, 3800117], ...
%p A254758 with(orthopoly): a := n -> `if`(n=0,1, U(n,3)+17*U(n-1, 3)):
%p A254758 seq(a(n), n=0..19); # _Peter Luschny_, Feb 07 2015
%t A254758 LinearRecurrence[{6, -1}, {1, 23}, 20] (* _Jean-François Alcover_, Jun 28 2019 *)
%o A254758 (PARI) Vec((1+17*x)/(1-6*x+x^2) + O(x^30)) \\ _Michel Marcus_, Feb 08 2015
%Y A254758 Cf. A254759, A254757, A220414, A001653, A002315, A049310.
%K A254758 nonn,easy
%O A254758 0,2
%A A254758 _Wolfdieter Lang_, Feb 07 2015