This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027903 #26 Aug 29 2025 12:44:59
%S A027903 0,8,42,120,260,480,798,1232,1800,2520,3410,4488,5772,7280,9030,11040,
%T A027903 13328,15912,18810,22040,25620,29568,33902,38640,43800,49400,55458,
%U A027903 61992,69020,76560,84630,93248,102432,112200,122570,133560,145188,157472,170430
%N A027903 a(n) = n*(n + 1)*(3*n + 1).
%H A027903 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A027903 From _Wesley Ivan Hurt_, Dec 02 2013: (Start)
%F A027903 a(n) = n*(n + 1)*(3*n + 1).
%F A027903 a(n) = 3*n^3 + 4*n^2 + n.
%F A027903 a(n) = A002378(n) * A016777(n).
%F A027903 a(n) = A049451(n) * A001477(n+1).
%F A027903 a(n) = A001477(n) * A000567(n-1).
%F A027903 a(n) = A001477(n) * A001477(n+1) * A016777(n).
%F A027903 a(n) = A117642(n) + A016742(n) + A001477(n). (End)
%F A027903 From _Amiram Eldar_, Aug 15 2025: (Start)
%F A027903 Sum_{n>=1} 1/a(n) = 4 - sqrt(3)*Pi/4 - 9*log(3)/4.
%F A027903 Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/2 + 2*log(2) - 4. (End)
%F A027903 From _Elmo R. Oliveira_, Aug 29 2025: (Start)
%F A027903 G.f.: 2*x*(4 + 5*x)/(1 - x)^4.
%F A027903 E.g.f.: x*(8 + 13*x + 3*x^2)*exp(x).
%F A027903 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%p A027903 A027903:=n->n*(n + 1)*(3*n + 1); seq(A027903(n), n=0..100); # _Wesley Ivan Hurt_, Dec 02 2013
%t A027903 Table[n (n + 1) (3*n + 1), {n, 0, 100}] (* _Wesley Ivan Hurt_, Dec 02 2013 *)
%Y A027903 Cf. A000567, A001477, A002378, A016742, A016777, A049451, A117642.
%K A027903 nonn,easy,changed
%O A027903 0,2
%A A027903 _N. J. A. Sloane_