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a(n) = 3*n^3.
13

%I #50 Mar 02 2024 08:00:47

%S 0,3,24,81,192,375,648,1029,1536,2187,3000,3993,5184,6591,8232,10125,

%T 12288,14739,17496,20577,24000,27783,31944,36501,41472,46875,52728,

%U 59049,65856,73167,81000,89373,98304,107811,117912,128625,139968,151959,164616

%N a(n) = 3*n^3.

%H Vincenzo Librandi, <a href="/A117642/b117642.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n+1) = a(n) + 9*n^2 + 9*n + 3 with a(0) = 0. - _Jean-Bernard François_, Oct 04 2013

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, May 27 2021

%F From _Amiram Eldar_, Jan 10 2023: (Start)

%F Sum_{n>=1} 1/a(n) = zeta(3)/3.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = zeta(3)/4. (End)

%F a(n) = n*(2*A000217(n) + A000217(2*n-1)). - _Charlie Marion_, Mar 01 2024

%p seq(3*n^3, n=0..38); # _Nathaniel Johnston_, Jun 26 2011

%t 3*Range[0, 35]^3 (* _Alonso del Arte_, Oct 04 2013 *)

%o (Magma) [3*n^3: n in [0..40]]; // _Vincenzo Librandi_, Jun 26 2011

%o (PARI) a(n)=3*n^3 \\ _Charles R Greathouse IV_, Oct 12 2017

%Y Cf. A000578, A002117, A033431.

%K nonn,easy

%O 0,2

%A _Roger L. Bagula_, Apr 10 2006

%E Edited by _N. J. A. Sloane_, Apr 30 2006