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 A154355 #58 May 15 2025 21:34:34
%S A154355 13,2,41,130,269,458,697,986,1325,1714,2153,2642,3181,3770,4409,5098,
%T A154355 5837,6626,7465,8354,9293,10282,11321,12410,13549,14738,15977,17266,
%U A154355 18605,19994,21433,22922,24461,26050,27689,29378
%N A154355 a(n) = 25*n^2 - 36*n + 13.
%C A154355 The identity (1250*n^2 - 1800*n + 649)^2 - (25*n^2 - 36*n + 13)*(250*n - 180)^2 = 1 can be written as A154358(n)^2 - a(n)*A154360(n)^2 = 1. See also the third comment in A154357.
%C A154355 Numbers of the form (3n-2)^2 + (4n-3)^2. - _Bruno Berselli_, Dec 12 2011
%C A154355 From _Klaus Purath_, May 06 2025: (Start)
%C A154355 25*a(n)-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 A154355 All a(n) = D satisfy the Pell equation (k*x)^2 - D*(5*y)^2 = -1 for any integer n where a(1-n) = A154357(n). The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*5^2 - 1), x(0) = 1, x(1) = 4*D*5^2 - 1, y(0) = 1, y(1) = 4*D*5^2 - 3. The two recurrences are of the form (4*D*5^2 - 2, -1).
%C A154355 It follows from the above that the terms of this sequence and of A154357 belong to A031396. (End)
%H A154355 Vincenzo Librandi, <a href="/A154355/b154355.txt">Table of n, a(n) for n = 0..10000</a>
%H A154355 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A154355 a(n) = A007533(n-1), n>0. - _R. J. Mathar_, Jan 14 2009
%F A154355 G.f.: (13 - 37*x + 74*x^2) / (1-x)^3. - _R. J. Mathar_, Jan 05 2011
%F A154355 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Feb 21 2012
%F A154355 E.g.f.: (13 - 11*x + 25*x^2) * exp(x). - _G. C. Greubel_, Sep 14 2016
%F A154355 From _Klaus Purath_, May 06 2025: (Start)
%F A154355 a(n) = (3*n-2)^2 + (4*n-3)^2.
%F A154355 25*a(n) - 1 = (25*n - 18)^2. (End)
%t A154355 Table[25n^2-36n+13,{n,0,40}] (* _Harvey P. Dale_, Apr 02 2011 *)
%t A154355 LinearRecurrence[{3, -3, 1}, {13, 2, 41}, 50] (* _Vincenzo Librandi_, Feb 21 2012 *)
%o A154355 (PARI) for(n=0, 40, print1(25*n^2 - 36*n + 13", ")); \\ _Vincenzo Librandi_, Feb 21 2012
%o A154355 (Magma) [25*n^2-36*n+13: n in [0..40]]; // _Bruno Berselli_, Sep 15 2016
%Y A154355 Cf. A154354, A154358, A154359, A154360, A154361, A202141.
%Y A154355 Essentially a duplicate of A007533.
%K A154355 nonn,easy
%O A154355 0,1
%A A154355 _Vincenzo Librandi_, Jan 07 2009
%E A154355 Offset corrected from _R. J. Mathar_, Jan 05 2011
%E A154355 First comment rewritten by _Bruno Berselli_, Dec 12 2011