A268484 - OEIS

%I #29 Nov 16 2024 21:30:34

%S 3,18,53,116,215,358,553,808,1131,1530,2013,2588,3263,4046,4945,5968,

%T 7123,8418,9861,11460,13223,15158,17273,19576,22075,24778,27693,30828,

%U 34191,37790,41633,45728,50083,54706,59605,64788,70263,76038,82121,88520,95243

%N a(n) = (n + 1)*(4*n^2 + 14*n + 9)/3.

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

%F G.f.: (3 + 6*x - x^2)/(x - 1)^4.

%F a(n) = Sum_{k = 0..n} (2*k + 1)*(2*k + 3) = Sum_{k = 0..n} A005408(k)*A005408(k + 1).

%F Sum_{n>=0} 1/a(n) = 0.4315109123788144393864...

%e a(0) = 1*3 = 3;

%e a(1) = 1*3 + 3*5 = 18;

%e a(2) = 1*3 + 3*5 + 5*7 = 53;

%e a(3) = 1*3 + 3*5 + 5*7 + 7*9 = 116, etc.

%t Table[(n + 1) ((4 n^2 + 14 n + 9)/3), {n, 0, 40}]

%t LinearRecurrence[{4, -6, 4, -1}, {3, 18, 53, 116}, 40]

%o (PARI) a(n)=(n+1)*(4*n^2+14*n+9)/3 \\ _Charles R Greathouse IV_, Jul 26 2016

%Y Cf. A000466, A005408, A007290, A135036.

%K nonn,easy

%O 0,1

%A _Ilya Gutkovskiy_, Feb 12 2016