A173707 Partial sums of floor(n^3/3).

0, 0, 2, 11, 32, 73, 145, 259, 429, 672, 1005, 1448, 2024, 2756, 3670, 4795, 6160, 7797, 9741, 12027, 14693, 17780, 21329, 25384, 29992, 35200, 41058, 47619, 54936, 63065, 72065, 81995, 92917, 104896, 117997, 132288, 147840, 164724, 183014, 202787, 224120, 247093, 271789, 298291, 326685, 357060, 389505, 424112, 460976, 500192, 541858

Offset

0,3

Comments

Partial sums of A131476.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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,-6,5,-5,6,-4,1).

Formula

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

a(n) = round((n^4 + 2*n^3 + n^2 - 4*n)/12).

a(n) = round((n^4 + 2*n^3 + n^2 - 4*n - 2)/12).

a(n) = floor((n^4 + 2*n^3 + n^2 - 4*n)/12).

a(n) = ceiling((n+1)*(n^3 + n^2 - 4)/12).

a(n) = a(n-3) + n^3 - 3*n^2 + 5*n - 4, n > 2.

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

G.f.: x^2*(2 + 3*x + x^3) / ( (1+x+x^2)*(1-x)^5 ).

a(n) = n^4/12 + n^3/6 + n^2/12 - n/3 - 1/9 + A061347(n+1)/9. (End)

Example

a(4) = floor(1/3) + floor(8/3) + floor(27/3) + floor(64/3) = 32.

Maple

A061347 := proc(n) op(1+(n mod 3), [-2, 1, 1]) ; end proc:
A173707 := proc(n) n^4/12+n^3/6+n^2/12-n/3-1/9 ; %+A061347(n+1)/9 ; end proc:
# program replaced by a structured version by R. J. Mathar, Nov 26 2010

Mathematica

Table[Sum[Floor[k^3/3], {k, 0, n}], {n, 0, 60}] (* G. C. Greubel, Nov 23 2016 *)
Accumulate[Table[Floor[n^3/3], {n, 0, 60}]] (* or *) LinearRecurrence[{4, -6, 5, -5, 6, -4, 1}, {0, 0, 2, 11, 32, 73, 145}, 60] (* Harvey P. Dale, May 29 2018 *)

Prog

(Magma) [Floor((n^4+2*n^3+n^2-4*n)/12): n in [0..60]]; // Vincenzo Librandi, May 08 2011
(PARI) a(n)=(n^4+2*n^3+n^2-4*n)\12 \\ Charles R Greathouse IV, May 08 2011
(Sage) [floor(n*(n^3 +2*n^2 +n -4)/12) for n in (0..60)] # G. C. Greubel, Jul 02 2019
(Python)
def A173707(n): return n*(n*(n*(n + 2) + 1) - 4)//12 # Chai Wah Wu, Feb 02 2023

Crossrefs

Cf. A131476.

Sequence in context: A033994 A023659 A094792 * A332612 A192347 A031400

Adjacent sequences: A173704 A173705 A173706 * A173708 A173709 A173710

Keyword

nonn,easy

Author

Mircea Merca, Nov 25 2010

Status

approved