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 A157757 #29 Apr 09 2025 22:17:18
%S A157757 89,3898,13325,28370,49033,75314,107213,144730,187865,236618,290989,
%T A157757 350978,416585,487810,564653,647114,735193,828890,928205,1033138,
%U A157757 1143689,1259858,1381645,1509050,1642073,1780714,1924973,2074850
%N A157757 a(n) = 2809*n^2 - 4618*n + 1898.
%C A157757 The identity (15780962*n^2-25943924*n+10662963)^2-(2809*n^2-4618*n+1898)*(297754*n-244754)^2=1 can be written as A157759(n)^2-a(n)*A157758(n)^2=1.
%C A157757 From _Klaus Purath_, Mar 31 2025: (Start)
%C A157757 Numbers k such that k*53^2-1 is a square, and k is the sum of two squares (see FORMULA).
%C A157757 All a(n) = D satisfy the Pell equation (k*x)^2 - D*(53*y)^2 = -1 for any integer n where a(1-n) = A157760(n). The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*53^2 - 1), x(0) = 1, x(1) = 4*D*53^2 - 1, y(0) = 1, y(1) = 4*D*53^2 - 3. The two recurrences are of the form (4*D*53^2 - 2, -1).
%C A157757 It follows from the above that this sequence and A157760 are subsequences of A031396. (End)
%H A157757 Vincenzo Librandi, <a href="/A157757/b157757.txt">Table of n, a(n) for n = 1..10000</a>
%H A157757 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A157757 a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
%F A157757 G.f.: x*(-89-3631*x-1898*x^2)/(x-1)^3.
%F A157757 a(n) = (28*n - 23)^2 + (45*n - 37)^2. - _Klaus Purath_, Mar 31 2025
%F A157757 53^2*a(n) - 1 = (2809*n-2309)^2. - _Klaus Purath_, Mar 31 2025
%t A157757 LinearRecurrence[{3,-3,1},{89,3898,13325},40]
%t A157757 Table[2809n^2-4618n+1898,{n,40}] (* _Harvey P. Dale_, Aug 02 2024 *)
%o A157757 (Magma) I:=[89, 3898, 13325]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
%o A157757 (PARI) a(n) = 2809*n^2 - 4618*n + 1898;
%Y A157757 Subsequence of A031396.
%Y A157757 Cf. A157758, A157759.
%K A157757 nonn,easy
%O A157757 1,1
%A A157757 _Vincenzo Librandi_, Mar 06 2009