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A269232 a(n) = (n + 1)*(6*n^2 + 15*n + 4)/2.

%I A269232 #20 Sep 08 2022 08:46:15
%S A269232 2,25,87,206,400,687,1085,1612,2286,3125,4147,5370,6812,8491,10425,
%T A269232 12632,15130,17937,21071,24550,28392,32615,37237,42276,47750,53677,
%U A269232 60075,66962,74356,82275,90737,99760,109362,119561,130375,141822,153920,166687,180141
%N A269232 a(n) = (n + 1)*(6*n^2 + 15*n + 4)/2.
%H A269232 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A269232 G.f.: (2 + 17*x - x^2)/(x - 1)^4.
%F A269232 a(n) = Sum_{k=0..n} (3*k + (3*k+1)*(3*k+2)) = Sum_{k=0..n} (A008585(k) + A001504(k)).
%F A269232 Sum_{n>=0} 1/a(n) = 0.56407113696623548787861365289...
%e A269232 a(0) = 0 + 1*2 = 2;
%e A269232 a(1) = 0 + 1*2 + 3 + 4*5 = 25;
%e A269232 a(2) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 = 87;
%e A269232 a(3) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;
%e A269232 a(4) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + 13*14 = 400, etc.
%t A269232 Table[(n + 1) ((6 n^2 + 15 n + 4)/2), {n, 0, 38}]
%t A269232 Table[Sum[3 k + (3 k + 1) (3 k + 2), {k, 0, n}], {n, 0, 38}]
%t A269232 LinearRecurrence[{4, -6, 4, -1}, {2, 25, 87, 206}, 39]
%o A269232 (PARI) Vec((2 + 17*x - x^2)/(x - 1)^4 + O(x^50)) \\ _Michel Marcus_, Feb 22 2016
%o A269232 (Magma) [(n+1)*(6*n^2+15*n+4)/2: n in [0..40]]; // _Vincenzo Librandi_, Feb 22 2016
%Y A269232 Cf. A001477, A001504, A008585.
%K A269232 nonn,easy
%O A269232 0,1
%A A269232 _Ilya Gutkovskiy_, Feb 20 2016