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 A275793 #26 Sep 08 2022 08:46:17
%S A275793 9,43,249,1451,8457,49291,287289,1674443,9759369,56881771,331531257,
%T A275793 1932305771,11262303369,65641514443,382586783289,2229879185291,
%U A275793 12996688328457,75750250785451,441504816384249,2573278647520043,14998167068736009,87415723764896011
%N A275793 The x members of the positive proper solutions (x = x1(n), y = y1(n)) of the first class for the Pell equation x^2 - 2*y^2 = +7^2.
%C A275793 This gives the (increasingly sorted) positive x members of the first class of the proper solutions (x1(n), y1(n)) to the Pell equation x^2 - 2*y^2 = +7^2. For the y1(n) solutions see 2*A275794(n). The solutions for the second class (x2(n), y2(n)) are found in A275795(n) and 2*A275796(n).
%C A275793 All solutions, including the improper ones, are given in A106525(n) and 2*A276600(n+2).
%C A275793 See also the comments on A263012 which apply here mutatis mutandis.
%C A275793 This is for the Pell equations x^2 - 2*y^2 = z^2, besides z^2 = 1 the first instance with proper solutions. For z^2 > 1 there seem to be always two classes of such solutions. For z^2 = 1 there is only one class of proper solutions. These z^2 values seem to appear for z from A058529 (prime factors are +1 or -1 (mod 8)).
%D A275793 T. Nagell, Introduction to Number Theory, Wiley, 1951, Theorem 109, pp. 207-208.
%H A275793 Colin Barker, <a href="/A275793/b275793.txt">Table of n, a(n) for n = 0..1000</a>
%H A275793 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A275793 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F A275793 a(n) = 43*S(n-1, 6) - 9*S(n-2, 6), with the Chebyshev polynomials S(n, 6) = A001109(n+1), n >= -1, with S(-2, 6) = -1.
%F A275793 O.g.f: (9 - 11*x)/(1 - 6*x + x^2).
%F A275793 a(n) = 6*a(n-1) - a(n-2) for n >= 1, with a(-1) = 11 and a(0) = 9.
%F A275793 a(n) = (((3-2*sqrt(2))^n*(-8+9*sqrt(2))+(3+2*sqrt(2))^n*(8+9*sqrt(2)))) / (2*sqrt(2)). - _Colin Barker_, Sep 28 2016
%F A275793 a(n) = 9*A000129(2*n+1) - A000129(2*n). - _G. C. Greubel_, Sep 15 2021
%e A275793 The first positive proper fundamental solution (x = x1(n), y = y1(n)) of x^2 - 2*y^2 = 49 are [9, 4], [43, 30], [249, 176], [1451, 1026], [8457, 5980], [49291, 34854], [287289, 203144], [1674443, 1184010], ...
%e A275793 The first positive proper fundamental solution of the second class (x = x2(n), y = y2(n)) are [11, 6], [57, 40], [331, 234], [1929, 1364], [11243, 7950], [65529, 46336], [381931, 270066], [2226057, 1574060], ...
%t A275793 RecurrenceTable[{a[n]== 6a[n-1] -a[n-2], a[-1]==11, a[0]==9}, a, {n,0,25}] (* _Michael De Vlieger_, Sep 28 2016 *)
%t A275793 Table[9*Fibonacci[2*n+1, 2] - Fibonacci[2*n, 2], {n,0,30}] (* _G. C. Greubel_, Sep 15 2021 *)
%o A275793 (PARI) a(n) = round((((3-2*sqrt(2))^n*(-8+9*sqrt(2))+(3+2*sqrt(2))^n*(8+9*sqrt(2)))) / (2*sqrt(2))) \\ _Colin Barker_, Sep 28 2016
%o A275793 (PARI) Vec((9-11*x)/(1-6*x+x^2) + O(x^30)) \\ _Colin Barker_, Oct 02 2016
%o A275793 (Magma) I:=[9,43]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..31]]; // _G. C. Greubel_, Sep 15 2021
%o A275793 (Sage)
%o A275793 def P(n): return lucas_number1(n, 2, -1);
%o A275793 [9*P(2*n+1) - P(2*n) for n in (0..30)] # _G. C. Greubel_, Sep 15 2021
%Y A275793 Cf. A000129, A001109, A058529, A106525.
%Y A275793 Cf. A263012, A275794, A275795, A275796, 2*A276600.
%K A275793 nonn,easy
%O A275793 0,1
%A A275793 _Wolfdieter Lang_, Sep 27 2016