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 A117748 Triangular numbers divisible by 3. 3
 0, 3, 6, 15, 21, 36, 45, 66, 78, 105, 120, 153, 171, 210, 231, 276, 300, 351, 378, 435, 465, 528, 561, 630, 666, 741, 780, 861, 903, 990, 1035, 1128, 1176, 1275, 1326, 1431, 1485, 1596, 1653, 1770, 1830, 1953, 2016, 2145, 2211, 2346, 2415, 2556, 2628, 2775 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Colin Barker, Table of n, a(n) for n = 1..1000 D. H. Lehmer, Recurrence formulas for certain divisor functions, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156. Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). FORMULA a(n) = 3*A001318(n-1). - Michel Marcus, Apr 24 2016 From Colin Barker, Apr 24 2016: (Start) a(n) = 3*(1-(-1)^n + 2*(-3+(-1)^n)*n + 6*n^2)/16. a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. G.f.: 3*x^2*(1+x+x^2) / ((1-x)^3*(1+x)^2). (End) E.g.f.: 3*(-1 - 2*x + exp(2*x) + 6*x^2*exp(2*x))*exp(-x)/16. - Ilya Gutkovskiy, Apr 24 2016 a(n) = A299412(n)/A007494(n). - Justin Gaetano, Feb 15 2018 Sum_{n>=2} 1/a(n) = 2 - 2*Pi/(3*sqrt(3)). - Amiram Eldar, Mar 24 2021 MATHEMATICA LinearRecurrence[{1, 2, -2, -1, 1}, {0, 3, 6, 15, 21}, 50] (* G. C. Greubel, Jun 19 2017 *) Select[Accumulate[Range[0, 100]], Divisible[#, 3]&] (* Harvey P. Dale, Feb 11 2018 *) PROG (PARI) lista(nn) = {for (i = 0, nn, t = i*(i+1)/2; if (t % 3 == 0, print1(t, ", "); )); } \\ Michel Marcus, Jun 01 2013 (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 CROSSREFS Cf. A000217, A001318, A007494, A299412.. Sequence in context: A095869 A292610 A069559 * A061066 A093799 A087359 Adjacent sequences: A117745 A117746 A117747 * A117749 A117750 A117751 KEYWORD easy,nonn AUTHOR Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006 STATUS approved

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