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A164578
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Integers of the form (k+1)*(2k+1)/12.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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)
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MATHEMATICA
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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 *)
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PROG
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(PARI) 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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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