%I #20 Sep 08 2022 08:46:15

%S 2,25,87,206,400,687,1085,1612,2286,3125,4147,5370,6812,8491,10425,

%T 12632,15130,17937,21071,24550,28392,32615,37237,42276,47750,53677,

%U 60075,66962,74356,82275,90737,99760,109362,119561,130375,141822,153920,166687,180141

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: (2 + 17*x - x^2)/(x - 1)^4.

%F a(n) = Sum_{k=0..n} (3*k + (3*k+1)*(3*k+2)) = Sum_{k=0..n} (A008585(k) + A001504(k)).

%F Sum_{n>=0} 1/a(n) = 0.56407113696623548787861365289...

%e a(0) = 0 + 1*2 = 2;

%e a(1) = 0 + 1*2 + 3 + 4*5 = 25;

%e a(2) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 = 87;

%e a(3) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 = 206;

%e a(4) = 0 + 1*2 + 3 + 4*5 + 6 + 7*8 + 9 + 10*11 + 12 + 13*14 = 400, etc.

%t Table[(n + 1) ((6 n^2 + 15 n + 4)/2), {n, 0, 38}]

%t Table[Sum[3 k + (3 k + 1) (3 k + 2), {k, 0, n}], {n, 0, 38}]

%t LinearRecurrence[{4, -6, 4, -1}, {2, 25, 87, 206}, 39]

%o (PARI) Vec((2 + 17*x - x^2)/(x - 1)^4 + O(x^50)) \\ _Michel Marcus_, Feb 22 2016

%o (Magma) [(n+1)*(6*n^2+15*n+4)/2: n in [0..40]]; // _Vincenzo Librandi_, Feb 22 2016

%Y Cf. A001477, A001504, A008585.

%K nonn,easy

%O 0,1

%A _Ilya Gutkovskiy_, Feb 20 2016