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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.
9

%I #32 Sep 08 2022 08:46:00

%S 0,10,990,97020,9506980,931587030,91286021970,8945098566040,

%T 876528373449960,85890835499530050,8416425350580494950,

%U 824723793521388975060,80814515339745539060940,7918997779501541438997070,775980967875811315482651930,76038215854050007375860892080

%N 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 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 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 Colin Barker, <a href="/A200993/b200993.txt">Table of n, a(n) for n = 0..500</a>

%H 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 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 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 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).

%F G.f. 10*x / ((1-x)*(x^2-98*x+1)). - _R. J. Mathar_, Dec 20 2011

%F a(n) = 99*a(n-1)-99*a(n-2)+a(n-3) for n>2. - _Colin Barker_, Mar 02 2016

%F 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 3*0 = 2*0.

%e 3*10 = 2*15.

%e 3*990 = 2*1485.

%e 3*97020 = 2*145530.

%t LinearRecurrence[{99,-99,1},{0,10,990},20] (* _Harvey P. Dale_, Feb 25 2018 *)

%o (PARI) concat(0, Vec(10*x/((1-x)*(1-98*x+x^2)) + O(x^40))) \\ _Colin Barker_, Mar 02 2016

%o (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 Cf. A001652, A029549, A053141, A075528, A200994-A201008.

%K nonn,easy

%O 0,2

%A _Charlie Marion_, Dec 15 2011