A164578 -id:A164578 - cp's OEIS frontend

A164579 Integer averages of halves of first cubes of natural numbers (n^3)/2 for some n.

6, 56, 81, 198, 480, 578, 950, 1656, 1875, 2646, 3968, 4356, 5670, 7800, 8405, 10406, 13536, 14406, 17238, 21560, 22743, 26550, 32256, 33800, 38726, 46008, 47961, 54150, 63200, 65610, 73206, 84216, 87131, 96278, 109440, 112908, 123750, 139256

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Also, integers of the form (1/8)*n*(n+1)^2 for some n. - Zak Seidov, Aug 17 2009

			1/2, 9/4, 6, 25/2, 45/2, 147/4, 56, 81, ...

Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A050248, A051456, A078617, A078618, A136116, A154293, A164576, A164577, A164578

s=0;lst={};Do[a=(s+=(n^3)/2)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,3*5!}];lst
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 *)
Module[{nn=200,ac},ac=Accumulate[Range[nn]^3/2];Select[#[[1]]/#[[2]]&/@ Thread[{ac,Range[nn]}],IntegerQ]] (* Harvey P. Dale, Jan 28 2020 *)

forstep(n=3, 150, [4,1,3], print1(n*(n+1)^2>>3, ", ")); \\ Charles R Greathouse IV, Nov 02 2009

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

A164619 Integers of the form A164577(k)/3.

Original entry on oeis.org

4, 15, 54, 75, 132, 169, 320, 459, 735, 847, 1104, 1250, 1764, 2175, 2904, 3179, 3780, 4107, 5200, 6027, 7425, 7935, 9024, 9604, 11492, 12879, 15162, 15979, 17700, 18605, 21504, 23595, 26979, 28175, 30672, 31974, 36100, 39039, 43740, 45387, 48804

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

The sequence members are the third of the average of a set of smallest cubes, if integer.

			A third of the average of the first cube, A164577(1)/3=1/3, is not an integer and does not contribute to the sequence.
A third of the average of the first two cubes, A164577(2)/3=4, is an integer and defines a(1)=4 of the sequence.

Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1,2,-4,2,2,-4,2,-1,2,-1,-1,2,-1).

Cf. A050248, A051456, A078617, A078618, A136116, A154293, A164286, A164576, A164577, A164578, A164579.

s=0;lst={};Do[a=(s+=(n^3)/3)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,2*5!}]; lst
LinearRecurrence[{2,-1,-1,2,-1,2,-4,2,2,-4,2,-1,2,-1,-1,2,-1},{4,15,54,75,132,169,320,459,735,847,1104,1250,1764,2175,2904,3179,3780},50] (* Harvey P. Dale, Apr 06 2016 *)

Vec(x*(x^14 +x^13 +16*x^12 +10*x^11 +47*x^10 -22*x^9 +61*x^8 +10*x^7 +88*x^6 +8*x^5 +43*x^4 -14*x^3 +28*x^2 +7*x +4) / ((x -1)^4*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^2) + O(x^100)) \\ Colin Barker, Oct 27 2014

a(n) = +2*a(n-1) -a(n-2) -a(n-3) +2*a(n-4) -a(n-5) +2*a(n-6) -4*a(n-7) +2*a(n-8) +2*a(n-9) -4*a(n-10) +2*a(n-11) -a(n-12) +2*a(n-13) -a(n-14) -a(n-15) +2*a(n-16) -a(n-17). - R. J. Mathar, Jan 25 2011

G.f.: x*(x^14 +x^13 +16*x^12 +10*x^11 +47*x^10 -22*x^9 +61*x^8 +10*x^7 +88*x^6 +8*x^5 +43*x^4 -14*x^3 +28*x^2 +7*x +4) / ((x -1)^4*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^2). - Colin Barker, Oct 27 2014

Edited by R. J. Mathar, Aug 20 2009

cp's OEIS Frontend

A164579 Integer averages of halves of first cubes of natural numbers (n^3)/2 for some n.

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