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A200994 Triangular numbers, T(m), that are three-halves of another triangular number; T(m) such that 2*T(m) = 3*T(k) for some k.

%I A200994 #27 Sep 08 2022 08:46:00
%S A200994 0,15,1485,145530,14260470,1397380545,136929032955,13417647849060,
%T A200994 1314792560174940,128836253249295075,12624638025870742425,
%U A200994 1237085690282083462590,121221773009618308591410,11878496669252312158495605,1163971451813716973223977895
%N A200994 Triangular numbers, T(m), that are three-halves of another triangular number; T(m) such that 2*T(m) = 3*T(k) for some k.
%C A200994 For n > 1, a(n) =  98*a(n-1) - a(n-2) + 15.  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).  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 A200994 Colin Barker, <a href="/A200994/b200994.txt">Table of n, a(n) for n = 0..500</a>
%H A200994 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 A200994 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 A200994 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 A200994 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 A200994 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).
%F A200994 From _Colin Barker_, Mar 02 2016: (Start)
%F A200994 a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n>2.
%F A200994 G.f.: 15*x / ((1-x)*(1-98*x+x^2)). (End)
%F A200994 a(n) = (-10+(5-2*sqrt(6))*(49+20*sqrt(6))^(-n)+(5+2*sqrt(6))*(49+20*sqrt(6))^n)/64. - _Colin Barker_, Mar 03 2016
%e A200994 2*0 = 3*0.
%e A200994 2*15 = 3*10.
%e A200994 2*1485 = 3*990.
%e A200994 2*145530 = 3*97020.
%t A200994 LinearRecurrence[{99, -99, 1}, {0, 15, 1485}, 20] (* _T. D. Noe_, Feb 15 2012 *)
%o A200994 (PARI) concat(0, Vec(15*x/((1-x)*(1-98*x+x^2)) + O(x^20))) \\ _Colin Barker_, Mar 02 2016
%o A200994 (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(15*x/((1-x)*(1-98*x+x^2)))); // _G. C. Greubel_, Jul 15 2018
%Y A200994 Cf. A001652, A029549, A053141, A075528, A200993-A201008.
%Y A200994 See also A352181, A352182.
%K A200994 nonn,easy
%O A200994 0,2
%A A200994 _Charlie Marion_, Feb 15 2012