A014817 a(n) = Sum_{k=1..n} floor(k^2/n).

1, 2, 4, 7, 9, 13, 18, 24, 29, 34, 42, 51, 57, 67, 78, 90, 97, 110, 122, 137, 149, 163, 180, 198, 211, 226, 246, 265, 281, 303, 324, 348, 365, 386, 412, 439, 457, 483, 512, 540, 561, 590, 618, 651, 679, 709, 742

Offset

1,2

References

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 103.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Formula

a(n) = n +A166375(n).

For prime p>2, a(p) = (p^2+2)/3 - A228131(p)/p. In particular, for prime p==1 (mod 4), a(p) = (p^2+2)/3. - Max Alekseyev, Aug 11 2013

Example

Row sums of the underlying triangle of floor(k^2/n), 1<=k<=n:
1;
0,2;
0,1,3;
0,1,2,4;
0,0,1,3,5;
0,0,1,2,4,6;
0,0,1,2,3,5,7;
0,0,1,2,3,4,6,8;
0,0,1,1,2,4,5,7,9;
0,0,0,1,2,3,4,6,8,10;
- R. J. Mathar, Aug 09 2013

Maple

A014817 := m->sum( floor(k^2/m), k=1..m);

Mathematica

Table[Sum[Floor[k^2/n], {k, n}], {n, 50}] (* Harvey P. Dale, Feb 23 2015 *)

Prog

(PARI) A014817(n)=sum(k=1, n, k^2\n)  \\ M. F. Hasler, Dec 11 2010
(PARI) a(n)=n^2-sum(m=1, n, sqrtint(n*m-1)) \\ Charles R Greathouse IV, Jun 20 2013
(Magma) [(&+[Floor(k^2/n): k in [1..n]]): n in [1..50]]; // G. C. Greubel, May 10 2018

Crossrefs

Cf. A177041, A166387, A166375, A165993, A227841, A227842.

Sequence in context: A036386 A280417 A099847 * A326502 A090893 A100486

Adjacent sequences: A014814 A014815 A014816 * A014818 A014819 A014820

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved