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A117748 Triangular numbers divisible by 3.

%I A117748 #56 Mar 24 2021 08:06:00
%S A117748 0,3,6,15,21,36,45,66,78,105,120,153,171,210,231,276,300,351,378,435,
%T A117748 465,528,561,630,666,741,780,861,903,990,1035,1128,1176,1275,1326,
%U A117748 1431,1485,1596,1653,1770,1830,1953,2016,2145,2211,2346,2415,2556,2628,2775
%N A117748 Triangular numbers divisible by 3.
%H A117748 Colin Barker, <a href="/A117748/b117748.txt">Table of n, a(n) for n = 1..1000</a>
%H A117748 D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0002-9904-1943-07880-9">Recurrence formulas for certain divisor functions</a>, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.
%H A117748 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A117748 a(n) = 3*A001318(n-1). - _Michel Marcus_, Apr 24 2016
%F A117748 From _Colin Barker_, Apr 24 2016: (Start)
%F A117748 a(n) = 3*(1-(-1)^n + 2*(-3+(-1)^n)*n + 6*n^2)/16.
%F A117748 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
%F A117748 G.f.: 3*x^2*(1+x+x^2) / ((1-x)^3*(1+x)^2).
%F A117748 (End)
%F A117748 E.g.f.: 3*(-1 - 2*x + exp(2*x) + 6*x^2*exp(2*x))*exp(-x)/16. - _Ilya Gutkovskiy_, Apr 24 2016
%F A117748 a(n) = A299412(n)/A007494(n). - _Justin Gaetano_, Feb 15 2018
%F A117748 Sum_{n>=2} 1/a(n) = 2 - 2*Pi/(3*sqrt(3)). - _Amiram Eldar_, Mar 24 2021
%t A117748 LinearRecurrence[{1,2,-2,-1,1}, {0, 3, 6, 15, 21}, 50] (* _G. C. Greubel_, Jun 19 2017 *)
%t A117748 Select[Accumulate[Range[0,100]],Divisible[#,3]&] (* _Harvey P. Dale_, Feb 11 2018 *)
%o A117748 (PARI) lista(nn) = {for (i = 0, nn, t = i*(i+1)/2; if (t % 3 == 0, print1(t, ", ");));} \\ _Michel Marcus_, Jun 01 2013
%o A117748 (PARI) concat(0, Vec(3*x^2*(1+x+x^2)/((1-x)^3*(1+x)^2) + O(x^50))) \\ _Colin Barker_, Apr 24 2016
%Y A117748 Cf. A000217, A001318, A007494, A299412..
%K A117748 easy,nonn
%O A117748 1,2
%A A117748 Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006