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A228888 a(n) = binomial(3*n + 2, 3).

%I A228888 #19 Dec 21 2023 11:27:01
%S A228888 10,56,165,364,680,1140,1771,2600,3654,4960,6545,8436,10660,13244,
%T A228888 16215,19600,23426,27720,32509,37820,43680,50116,57155,64824,73150,
%U A228888 82160,91881,102340,113564,125580,138415,152096,166650,182104,198485,215820,234136,253460
%N A228888 a(n) = binomial(3*n + 2, 3).
%H A228888 Vincenzo Librandi, <a href="/A228888/b228888.txt">Table of n, a(n) for n = 1..1000</a>
%H A228888 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F A228888 a(n) = binomial(3*n + 2, 3) = 1/6*(3*n)*(3*n + 1)*(3*n + 2).
%F A228888 a(-n) = - A006566(n).
%F A228888 a(n) = 1/6*A228889(n).
%F A228888 G.f.: (10*x + 16*x^2 + x^3)/(1 - x)^4 = 10*x + 56*x^2 + 165*x^3 + ....
%F A228888 Sum {n >= 1} 1/a(n) = 9/2 - 3/2*log(3) - 1/2*sqrt(3)*Pi.
%F A228888 Sum {n >= 1} (-1)^n/a(n) = 9/2 - 4*log(2) - 1/3*sqrt(3)*Pi.
%e A228888 From _Bruno Berselli_, Jun 26 2018: (Start)
%e A228888 Including 0, row sums of the triangle:
%e A228888 | 0|   .................................................................. 0
%e A228888 | 1|   2   3   4   ..................................................... 10
%e A228888 | 5|   6   7   8   9  10  11   ......................................... 56
%e A228888 |12|  13  14  15  16  17  18  19  20  21   ............................ 165
%e A228888 |22|  23  24  25  26  27  28  29  30  31  32  33  34   ................ 364
%e A228888 |35|  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50   .... 680
%e A228888 ...
%e A228888 in the first column of which we have the pentagonal numbers (A000326).
%e A228888 (End)
%p A228888 seq(binomial(3*n+2,3), n = 1..38);
%t A228888 Table[(Binomial[3 n + 2, 3]), {n, 1, 40}] (* _Vincenzo Librandi_, Sep 09 2013 *)
%o A228888 (Magma) [Binomial(3*n + 2, 3): n in [1..40]]; // _Vincenzo Librandi_, Sep 09 2013
%Y A228888 Cf. A006566 (binomial(3*n,3)) and A228887 (binomial(3*n + 1,3)).
%Y A228888 Cf. A228889.
%Y A228888 Similar sequences are listed in A316224.
%K A228888 nonn,easy
%O A228888 1,1
%A A228888 _Peter Bala_, Sep 09 2013