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%I #20 Sep 08 2022 08:46:00
%S 0,21,4095,794430,154115346,29897582715,5799976931385,
%T 1125165627105996,218276331681631860,42344483180609474865,
%U 8214611460706556491971,1593592278893891349967530,309148687493954215337208870,59973251781548223884068553271
%N Triangular numbers, T(m), that are three-quarters of another triangular number: T(m) such that 4*T(m)=3*T(k) for some k.
%C For n>1, a(n) = 194*a(n-1) - a (n-2) + 21. See A200993 for generalization.
%H Colin Barker, <a href="/A200998/b200998.txt">Table of n, a(n) for n = 0..425</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (195,-195,1).
%F G.f.: (21*x)/(1 - 195*x + 195*x^2 - x^3).
%F From _Colin Barker_, Mar 02 2016: (Start)
%F a(n) = 195*a(n-1)-195*a(n-2)+a(n-3) for n>2.
%F a(n) = ((97+56*sqrt(3))^(-n)*(-1+(97+56*sqrt(3))^n)*(-7+4*sqrt(3)+(7+4*sqrt(3))*(97+56*sqrt(3))^n))/128.
%F (End)
%e 4*0 = 3*0.
%e 4*21 = 3*28.
%e 4*4095 = 3*5640.
%e 4*794430 = 3*1059240.
%t LinearRecurrence[{195, -195, 1}, {0, 21, 4095}, 30] (* _Vincenzo Librandi_, Mar 03 2016 *)
%o (PARI) concat(0, Vec(21/(1 - 195*x + 195*x^2 - x^3) + O(x^99))) \\ _Charles R Greathouse IV_, Dec 20 2011
%o (Magma) I:=[0,21]; [n le 2 select I[n] else 194*Self(n-1) - Self(n-2) + 21: n in [1..20]]; // _Vincenzo Librandi_, Mar 03 2016
%Y Cf. A001652, A029549, A053141, A075528, A200993-A201008.
%K nonn,easy
%O 0,2
%A _Charlie Marion_, Dec 20 2011
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