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A173962 Averages of two consecutive odd cubes; a(n) = (n^3 + (n+2)^3)/2.

%I A173962 #17 Aug 24 2025 11:33:02
%S A173962 14,76,234,536,1030,1764,2786,4144,5886,8060,10714,13896,17654,22036,
%T A173962 27090,32864,39406,46764,54986,64120,74214,85316,97474,110736,125150,
%U A173962 140764,157626,175784,195286,216180,238514,262336,287694,314636,343210,373464,405446
%N A173962 Averages of two consecutive odd cubes; a(n) = (n^3 + (n+2)^3)/2.
%H A173962 Colin Barker, <a href="/A173962/b173962.txt">Table of n, a(n) for n = 1..1000</a>
%H A173962 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A173962 From _Colin Barker_, Jan 17 2015: (Start)
%F A173962 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A173962 G.f.: 2*x*(7*x^2+10*x+7)/(x-1)^4. (End)
%F A173962 From _Elmo R. Oliveira_, Aug 23 2025: (Start)
%F A173962 a(n) = 2*n*(4*n^2 + 3) = A271636(n)/2.
%F A173962 E.g.f.: 2*exp(x)*x*(7 + 12*x + 4*x^2). (End)
%e A173962 (1^3 + 3^3)/2 = 14, ...
%t A173962 f[n_]:=(n^3+(n+2)^3)/2;Table[f[n],{n,1,6!,2}]
%t A173962 Mean/@Partition[Range[1,81,2]^3,2,1] (* _Harvey P. Dale_, Apr 20 2015 *)
%o A173962 (PARI) Vec(2*x*(7*x^2+10*x+7)/(x-1)^4 + O(x^100)) \\ _Colin Barker_, Jan 17 2015
%Y A173962 Cf. A027575, A173960, A173961, A271636.
%K A173962 nonn,easy,changed
%O A173962 1,1
%A A173962 _Vladimir Joseph Stephan Orlovsky_, Mar 03 2010