A173704 Partial sums of floor(n^3/2).

0, 0, 4, 17, 49, 111, 219, 390, 646, 1010, 1510, 2175, 3039, 4137, 5509, 7196, 9244, 11700, 14616, 18045, 22045, 26675, 31999, 38082, 44994, 52806, 61594, 71435, 82411, 94605, 108105, 123000, 139384, 157352, 177004, 198441, 221769, 247095, 274531, 304190, 336190, 370650, 407694, 447447, 490039, 535601, 584269, 636180, 691476, 750300, 812800

Offset: 0

Author

Mircea Merca, Nov 25 2010

Keywords

• nonn

Comments

Partial sums of A036487.

Examples

			a(4) = floor(1/2) + floor(8/2) + floor(27/2) + floor(64/2) = 49.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Crossrefs

Cf. A036487.

Programs

Magma

[Round((n^4+2*n^3+n^2-2*n)/8): n in [0..40]]; // Vincenzo Librandi, Jun 22 2011

Maple

A173704 := proc(n) (n^4+2*n^3+n^2-2*n-1+(-1)^n)/8 ; end proc:

Mathematica

Table[Sum[Floor[k^3/2], {k, 0, n}], {n,0,50}] (* G. C. Greubel, Nov 23 2016 *)
Accumulate[Floor[Range[0,50]^3/2]] (* Harvey P. Dale, Jun 22 2025 *)

Formula

a(n) = Sum_{k=0..n} floor(k^3/2).

a(n) = round((n^4+2*n^3+n^2-2*n)/8).

a(n) = round((n^4+2*n^3+n^2-2*n-1)/8).

a(n) = floor((n^4+2*n^3+n^2-2*n)/8).

a(n) = ceiling((n-1)*(n+1)*(n^2+2*n+2)/8).

a(n) = a(n-2)+(n-1)*(2*n^2-n+2)/2, n>1.

From R. J. Mathar, Nov 26 2010: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).

G.f.: -x^2*(4+x+x^2) / ( (1+x)*(x-1)^5 ).

a(n) = (n^4 + 2*n^3 + n^2 - 2*n - 1 + (-1)^n)/8. (End)

Extensions

Maple program replaced by R. J. Mathar, Nov 26 2010