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%I #19 Apr 11 2024 10:11:03
%S 6,56,81,198,480,578,950,1656,1875,2646,3968,4356,5670,7800,8405,
%T 10406,13536,14406,17238,21560,22743,26550,32256,33800,38726,46008,
%U 47961,54150,63200,65610,73206,84216,87131,96278,109440,112908,123750,139256
%N Integer averages of halves of first cubes of natural numbers (n^3)/2 for some n.
%C Also, integers of the form (1/8)*n*(n+1)^2 for some n. - _Zak Seidov_, Aug 17 2009
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,-3,0,-3,3,0,1,-1).
%F G.f.: ( x*(6+50*x+25*x^2+99*x^3+132*x^4+23*x^5+39*x^6+10*x^7) ) / ( (1+x+x^2)^3*(x-1)^4 ). - _R. J. Mathar_, Jan 25 2011
%e 1/2, 9/4, 6, 25/2, 45/2, 147/4, 56, 81, ...
%t s=0;lst={};Do[a=(s+=(n^3)/2)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,3*5!}];lst
%t LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{6,56,81,198,480,578,950,1656,1875,2646},40] (* _Harvey P. Dale_, Jul 26 2017 *)
%t Module[{nn=200,ac},ac=Accumulate[Range[nn]^3/2];Select[#[[1]]/#[[2]]&/@ Thread[{ac,Range[nn]}],IntegerQ]] (* _Harvey P. Dale_, Jan 28 2020 *)
%o (PARI) forstep(n=3, 150, [4,1,3], print1(n*(n+1)^2>>3, ", ")); \\ _Charles R Greathouse IV_, Nov 02 2009
%Y Cf. A050248, A051456, A078617, A078618, A136116, A154293, A164576, A164577, A164578
%K nonn,easy
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Aug 16 2009
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