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 A157110 #29 Apr 23 2025 08:04:53
%S A157110 85,2522,8321,17482,30005,45890,65137,87746,113717,143050,175745,
%T A157110 211802,251221,294002,340145,389650,442517,498746,558337,621290,
%U A157110 687605,757282,830321,906722,986485,1069610,1156097,1245946,1339157,1435730,1535665
%N A157110 a(n) = 1681*n^2 - 2606*n + 1010.
%C A157110 The identity (5651522*n^2 - 8761372*n + 3395619)^2 - (1681*n^2 - 2606*n + 1010)*(137842*n - 106846)^2 = 1 can be written as A157112(n)^2 - a(n)*A157111(n)^2 = 1. - _Vincenzo Librandi_, Jan 25 2012
%C A157110 The continued fraction expansion of sqrt(a(n)) is [41n-32; {4, 1, 1, 4, 82n-64}]. - _Magus K. Chu_, Oct 03 2022
%C A157110 From _Klaus Purath_, Apr 18 2025: (Start)
%C A157110 a(n)*41^2-1 is a square, and a(n) is the sum of two squares (see FORMULA). There are no squares in this sequence. The odd prime factors of these terms are always of the form 4*k + 1.
%C A157110 All a(n) = D satisfy the Pell equation (k*x)^2 - D*(41*y)^2 = -1 for any integer n where a(1-n) = A157010(n). The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*41^2 - 1), x(0) = 1, x(1) = 4*D*41^2 - 1, y(0) = 1, y(1) = 4*D*41^2 - 3. The two recurrences are of the form (4*D*41^2 - 2, -1).
%C A157110 It follows from the above that the terms of this sequence and of A157010 belong to A031396. (End)
%H A157110 Vincenzo Librandi, <a href="/A157110/b157110.txt">Table of n, a(n) for n = 1..10000</a>
%H A157110 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&tstart=0">X^2-AY^2=1</a> [Broken link]
%H A157110 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A157110 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jan 25 2012
%F A157110 G.f.: x*(-85 - 2267*x - 1010*x^2)/(x-1)^3. - _Vincenzo Librandi_, Jan 25 2012
%F A157110 From _Klaus Purath_, Apr 18 2025: (Start)
%F A157110 a(n) = (9*n - 7)^2 + (40*n - 31)^2 for any integer n.
%F A157110 1681*a(n) - 1 = (1681*n - 1303)^2 for any integer n. (End)
%t A157110 LinearRecurrence[{3,-3,1},{85,2522,8321},40] (* _Vincenzo Librandi_, Jan 25 2012 *)
%t A157110 Table[1681*n^2-2606*n+1010,{n,40}] (* _Harvey P. Dale_, Nov 24 2024 *)
%o A157110 (Magma) I:=[85, 2522, 8321]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Jan 25 2012
%o A157110 (PARI) for(n=1, 22, print1(1681*n^2 - 2606*n + 1010", ")); \\ _Vincenzo Librandi_, Jan 25 2012
%Y A157110 Cf. A157111, A157112.
%K A157110 nonn,easy
%O A157110 1,1
%A A157110 _Vincenzo Librandi_, Feb 23 2009