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 A200993 #32 Sep 08 2022 08:46:00
%S A200993 0,10,990,97020,9506980,931587030,91286021970,8945098566040,
%T A200993 876528373449960,85890835499530050,8416425350580494950,
%U A200993 824723793521388975060,80814515339745539060940,7918997779501541438997070,775980967875811315482651930,76038215854050007375860892080
%N A200993 Triangular numbers, T(m), that are two-thirds of another triangular number; T(m) such that 3*T(m) = 2*T(k) for some k.
%C A200993 For n>1, a(n) = 98*a(n-1) - a (n-2) + 10. In general, for m>0, let b(n) be those triangular numbers such that for some triangular number c(n), (m+1)*b(n) = m*c(n). Then b(0) = 0, b(1)= A014105(m) and for n>1, b(n) = 2*A069129(m+1)*b(n-1) - b(n-2) + A014105(m).
%C A200993 Further, c(0) = 0, c(1) = A000384(m+1) and for n>1, c(n) = 2*A069129(m+1)*c(n-1) - c(n-2) + A000384(m+1).
%H A200993 Colin Barker, <a href="/A200993/b200993.txt">Table of n, a(n) for n = 0..500</a>
%H A200993 Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
%H A200993 Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_1.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
%H A200993 Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_2.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
%H A200993 Editors, L'Intermédiaire des Mathématiciens, <a href="/A072256/a072256_3.pdf">Query 4500: The equation x(x+1)/2 = y*(y+1)/3</a>, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
%H A200993 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).
%F A200993 G.f. 10*x / ((1-x)*(x^2-98*x+1)). - _R. J. Mathar_, Dec 20 2011
%F A200993 a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>2. - _Colin Barker_, Mar 02 2016
%F A200993 a(n) = (-10+(5-2*sqrt(6))*(49+20*sqrt(6))^(-n)+(5+2*sqrt(6))*(49+20*sqrt(6))^n)/96. - _Colin Barker_, Mar 07 2016
%e A200993 3*0 = 2*0.
%e A200993 3*10 = 2*15.
%e A200993 3*990 = 2*1485.
%e A200993 3*97020 = 2*145530.
%t A200993 LinearRecurrence[{99,-99,1},{0,10,990},20] (* _Harvey P. Dale_, Feb 25 2018 *)
%o A200993 (PARI) concat(0, Vec(10*x/((1-x)*(1-98*x+x^2)) + O(x^40))) \\ _Colin Barker_, Mar 02 2016
%o A200993 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(10*x/((1-x)*(x^2-98*x+1)))); // _G. C. Greubel_, Jul 15 2018
%Y A200993 Cf. A001652, A029549, A053141, A075528, A200994-A201008.
%K A200993 nonn,easy
%O A200993 0,2
%A A200993 _Charlie Marion_, Dec 15 2011