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 A201004 #34 Dec 02 2024 06:56:32
%S A201004 0,45,14535,4680270,1507032450,485259768675,156252138480945,
%T A201004 50312703331095660,16200534220474321620,5216521706289400466025,
%U A201004 1679703788890966475738475,540859403501184915787322970,174155048223592651917042257910,56077384668593332732371819724095
%N A201004 Triangular numbers, T(m), that are five-quarters of another triangular number; T(m) such that 4*T(m) = 5*T(k) for some k.
%H A201004 Vincenzo Librandi, <a href="/A201004/b201004.txt">Table of n, a(n) for n = 0..200</a>
%H A201004 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (323,-323,1).
%F A201004 For n > 1, a(n) = 322*a(n-1) - a(n-2) + 45. See A200994 for generalization.
%F A201004 G.f.: 45*x / ((1-x)*(x^2-322*x+1)). - _R. J. Mathar_, Aug 10 2014
%F A201004 From _Colin Barker_, Mar 02 2016: (Start)
%F A201004 a(n) = (-18 + (9-4*sqrt(5))*(161+72*sqrt(5))^(-n) + (9+4*sqrt(5))*(161+72*sqrt(5))^n)/128.
%F A201004 a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n > 2. (End)
%F A201004 a(n) = 45*A298271(n). - _Amiram Eldar_, Dec 01 2024
%e A201004 4*0 = 5*0.
%e A201004 4*45 = 5*36.
%e A201004 4*14535 = 5*11628.
%e A201004 4*4680270 = 5*3744216.
%t A201004 LinearRecurrence[{323, -323, 1}, {0, 45, 14535}, 20] (* _T. D. Noe_, Feb 15 2012 *)
%t A201004 CoefficientList[Series[-45 x/((x - 1) (x^2 - 322 x + 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 11 2014 *)
%o A201004 (PARI) concat(0, Vec(45*x/((1-x)*(1-322*x+x^2)) + O(x^15))) \\ _Colin Barker_, Mar 02 2016
%o A201004 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(45*x/((1-x)*(1-322*x+x^2)))); // _G. C. Greubel_, Jul 15 2018
%Y A201004 Cf. A001652, A029549, A053141, A075528, A200993-A201008, A298271.
%K A201004 nonn,easy
%O A201004 0,2
%A A201004 _Charlie Marion_, Feb 15 2012
%E A201004 a(7) corrected by _R. J. Mathar_, Aug 10 2014