A164577 -id:A164577 - cp's OEIS frontend

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

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

A164578 Integers of the form (k+1)*(2k+1)/12.

Original entry on oeis.org

10, 23, 65, 94, 168, 213, 319, 380, 518, 595, 765, 858, 1060, 1169, 1403, 1528, 1794, 1935, 2233, 2390, 2720, 2893, 3255, 3444, 3838, 4043, 4469, 4690, 5148, 5385, 5875, 6128, 6650, 6919, 7473, 7758, 8344, 8645, 9263, 9580, 10230, 10563, 11245, 11594

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

This can also be defined as integer averages of the first k halved squares, 1^2/2, 2^2/2, 3^2/2,... , 3^k/2, because sum_{j=1..k} j^2/2 = k*(k+1)*(2k+1)/12. The generating k are in A168489.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A078617, A078618, A154293, A164576, A164577.

s=0;lst={};Do[a=(s+=(n^2)/2)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,2*6!}];lst
Select[Table[((n+1)(2n+1))/12,{n,300}],IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{10,23,65,94,168},60] (* Harvey P. Dale, Jun 14 2017 *)

Vec(x*(10+13*x+22*x^2+3*x^3)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

a(n) = +a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5). G.f. x*(-10-13*x-22*x^2-3*x^3) / ((1+x)^2*(x-1)^3). - R. J. Mathar, Jan 25 2011
From Colin Barker, Jan 26 2016: (Start)
a(n) = (24*n^2+6*n-(-1)^n*(8*n+1)+1)/4.
a(n) = (12*n^2-n)/2 for n even.
a(n) = (12*n^2+7*n+1)/2 for n odd.
(End)

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

Original entry on oeis.org

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

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

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A164578 Integers of the form (k+1)*(2k+1)/12.

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

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Author

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Examples

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Programs

Mathematica

PARI

Formula