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A173961 Averages of two consecutive even cubes: (n^3 + (n+2)^3)/2.

%I A173961 #40 Aug 23 2025 14:45:00
%S A173961 4,36,140,364,756,1364,2236,3420,4964,6916,9324,12236,15700,19764,
%T A173961 24476,29884,36036,42980,50764,59436,69044,79636,91260,103964,117796,
%U A173961 132804,149036,166540,185364,205556,227164,250236,274820,300964,328716,358124,389236,422100
%N A173961 Averages of two consecutive even cubes: (n^3 + (n+2)^3)/2.
%H A173961 Vincenzo Librandi, <a href="/A173961/b173961.txt">Table of n, a(n) for n = 1..1000</a>
%H A173961 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A173961 G.f.: x*(4+20*x+20*x^2+4*x^3)/(1-4*x+6*x^2-4*x^3+x^4). - _Colin Barker_, Jan 04 2012
%F A173961 a(n) = 8*n^3 - 12*n^2 + 12*n - 4. - _Charles R Greathouse IV_, Jan 02 2012
%F A173961 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jul 02 2012
%F A173961 a(n) = 4*A005898(n-1).
%F A173961 E.g.f.: 4 + 4*exp(x)*(-1 + 2*x + 3*x^2 + 2*x^3). - _Elmo R. Oliveira_, Aug 23 2025
%e A173961 (0^3+2^3)/2=4, (2^3+4^3)/2=36, ...
%t A173961 f[n_]:=(n^3+(n+2)^3)/2;Table[f[n],{n,0,5!,2}]
%t A173961 CoefficientList[Series[(4+20*x+20*x^2+4*x^3)/(1-4*x+6*x^2-4*x^3+x^4),{x,0,40}],x] (* _Vincenzo Librandi_, Jul 02 2012 *)
%o A173961 (PARI) a(n)=4*n*(2*n^2-3*n+3)-4 \\ _Charles R Greathouse IV_, Jan 02 2012
%o A173961 (Magma) I:=[4, 36, 140, 364]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jul 02 2012
%Y A173961 Cf. A005898, A027575, A173960.
%K A173961 nonn,easy,changed
%O A173961 1,1
%A A173961 _Vladimir Joseph Stephan Orlovsky_, Mar 03 2010