A078617 Floor(average of first n squares).

1, 2, 4, 7, 11, 15, 20, 25, 31, 38, 46, 54, 63, 72, 82, 93, 105, 117, 130, 143, 157, 172, 188, 204, 221, 238, 256, 275, 295, 315, 336, 357, 379, 402, 426, 450, 475, 500, 526, 553, 581, 609, 638, 667, 697, 728, 760, 792, 825, 858, 892, 927, 963, 999, 1036, 1073

Offset: 1

Author

Joseph L. Pe, Dec 10 2002

Keywords

• easy
• nonn

Examples

			a(4) = floor((1 + 4 + 9 + 16)/4) = 7.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1)

Crossrefs

Cf. A000290, A000330, A171662.

Programs

Mathematica

Table[Floor[Mean[Range[n]^2]],{n,60}] (* Harvey P. Dale, Jun 19 2023 *)

PARI

{a(n) = n++; (2 * n^2 - n) \ 6} /* Michael Somos, Dec 14 2009 */

Formula

a(n) = floor((1/n)*(Sum_{i=1..n} i^2)) = floor( A000330(n)/n ).

a(n) = [(n + 1) * (2 * n + 1) / 6]. A171662(n) = a(-1 - n). - Michael Somos, Dec 14 2009

G.f.: -x*(1+x^3+x^4+x^2) / ( (1+x)*(1+x+x^2)*(x^2-x+1)*(x-1)^3 ). - R. J. Mathar, Feb 20 2011

a(n) = floor(n/(1-e^(-3/n))). Also see related exponential formula in A171662 with symmetry as above. - Richard R. Forberg, Jun 22 2013