A181286 Partial sums of floor(n^2/3) (A000212).

0, 0, 1, 4, 9, 17, 29, 45, 66, 93, 126, 166, 214, 270, 335, 410, 495, 591, 699, 819, 952, 1099, 1260, 1436, 1628, 1836, 2061, 2304, 2565, 2845, 3145, 3465, 3806, 4169, 4554, 4962, 5394, 5850, 6331, 6838, 7371, 7931, 8519, 9135, 9780, 10455, 11160, 11896

Offset

0,4

Comments

Column sums of:

1 4 9 16 25 36 49...

1 4 9 16...

1...

....................

--------------------

1 4 9 17 29 45 66...

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 (3,-3,2,-3,3,-1).

Formula

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

a(n) = round((2*n^3 + 3*n^2 - 3*n)/18).

a(n) = round((2*n^3 + 3*n^2 - 3*n - 2)/18).

a(n) = floor((2*n^3 + 3*n^2 - 3*n)/18).

a(n) = ceiling((2*n^3 + 3*n^2 - 3*n - 4)/18).

a(n) = a(n-3) + (n-1)^2, n > 2.

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

G.f.: x^2*(1+x)/((1+x+x^2)*(1-x)^4). - L. Edson Jeffery, Jan 16 2014

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6), n >= 6. - L. Edson Jeffery, Jan 16 2014

E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(-2 + x*(2 + x*(9 + 2*x))) + 6*cos(sqrt(3)*x/2) + 2*sqrt(3)*sin(sqrt(3)*x/2))/54. - Stefano Spezia, Oct 24 2022

Example

a(5) = 17 = 0 + 0 + 1 + 3 + 5 + 8.

Maple

a(n):=round((2*n^(3)+3*n^(2)-3*n)/(18))

Mathematica

Accumulate[Floor[Range[0, 80]^2/3]] (* Harvey P. Dale, Jun 14 2015 *)

Prog

(Magma) [Round((2*n^(3)+3*n^(2)-3*n)/(18)): n in [0..50]]; // Vincenzo Librandi, Jun 21 2011

Crossrefs

Cf. A000212.

Sequence in context: A008110 A008224 A036594 * A008138 A301123 A265047

Adjacent sequences: A181283 A181284 A181285 * A181287 A181288 A181289

Keyword

nonn,easy

Author

Mircea Merca, Oct 12 2010

Status

approved