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A294315 a(n) = 3*n^3 + n^2.

%I A294315 #27 Jan 10 2023 01:52:23
%S A294315 0,4,28,90,208,400,684,1078,1600,2268,3100,4114,5328,6760,8428,10350,
%T A294315 12544,15028,17820,20938,24400,28224,32428,37030,42048,47500,53404,
%U A294315 59778,66640,74008,81900,90334,99328,108900,119068,129850,141264,153328,166060,179478
%N A294315 a(n) = 3*n^3 + n^2.
%C A294315 All terms are even.
%H A294315 Muniru A Asiru, <a href="/A294315/b294315.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from Colin Barker)
%H A294315 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A294315 a(n) = 3*n^3 + n^2.
%F A294315 a(n) = A117642(n) + A000290(n).
%F A294315 a(n) = 2*A036659(n).
%F A294315 From _Colin Barker_, Dec 11 2017: (Start)
%F A294315 G.f.: 2*x*(2 + 6*x + x^2) / (1 - x)^4.
%F A294315 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.
%F A294315 (End)
%F A294315 From _Amiram Eldar_, Jan 10 2023: (Start)
%F A294315 Sum_{n>=1} 1/a(n) = Pi^2/6 + sqrt(3)*Pi/2 + 9*log(3)/2 - 9.
%F A294315 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - sqrt(3)*Pi - 6*log(2) + 9. (End)
%e A294315 a(3)=90 because 3*3^3 + 3^2 = 3*27 + 9 = 90.
%t A294315 Array[3 #^3 + #^2 &, 40, 0] (* or *)
%t A294315 LinearRecurrence[{4, -6, 4, -1}, {0, 4, 28, 90}, 40] (* or *)
%t A294315 CoefficientList[Series[2 x (2 + 6 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* _Michael De Vlieger_, Dec 12 2017 *)
%o A294315 (PARI) a(n) = 3*n^3 + n^2;
%o A294315 (PARI) concat(0, Vec(2*x*(2 + 6*x + x^2) / (1 - x)^4 + O(x^40))) \\ _Colin Barker_, Dec 11 2017
%o A294315 (GAP) A294315:=List([0..10^4],n -> 3 *n^3 + n^2 ); # _Muniru A Asiru_, Dec 11 2017
%Y A294315 Cf. A117642, A000290, A036659.
%K A294315 nonn,easy
%O A294315 0,2
%A A294315 _Jason Morgan_, Oct 28 2017