A181120 Partial sums of round(n^2/12) (A069905).

0, 0, 0, 1, 2, 4, 7, 11, 16, 23, 31, 41, 53, 67, 83, 102, 123, 147, 174, 204, 237, 274, 314, 358, 406, 458, 514, 575, 640, 710, 785, 865, 950, 1041, 1137, 1239, 1347, 1461, 1581, 1708, 1841, 1981, 2128, 2282, 2443, 2612, 2788, 2972, 3164, 3364, 3572

Offset

0,5

Comments

Number of triples of positive integers (a, b, c) such that 1 <= a <= b <= c and a + b + c <= n. - Leonhard Vogt, Apr 27 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..870

D. Barrera, M. J. Ibáñez, and S. Remogna, On the construction of trivariate near-best quasi-interpolants based on C^2 quartic splines on type-6 tetrahedral partitions, Journal of Computational and Applied, 2016, Volume 311, February 2017, Pages 252-261.

J. Brandts and A. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.

Jan Brandts and Apo Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.

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 (2,0,-1,-1,0,2,-1).

Formula

a(n) = round((2*n^3 + 3*n^2 - 6*n)/72).

a(n) = round((4*n^3 + 6*n^2 - 12*n - 7)/144).

a(n) = floor((2*n^3 + 3*n^2 - 6*n + 9)/72).

a(n) = ceiling((2*n^3 + 3*n^2 - 6*n + 9 - 16)/72).

a(n) = a(n-6) + (n^2 - 5*n + 8)/2, n > 5.

From R. J. Mathar, Oct 06 2010: (Start)

a(n) = (-1)^n/16 + n^3/36 - n^2/24 - n/12 + 7/144 - A049347(n)/9.

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

a(n) = A000601(n-3). - R. J. Mathar, Oct 11 2017

Example

a(5) = 4 = 0 + 0 + 0 + 1 + 1 + 2.

Maple

a:= n-> round(1/(72)*(2*n^(3)+3*n^(2)-6*n)): seq(a(n), n=0..50);

Prog

(PARI) a(n)=round(n*(2*n^2+3*n-6)/72) \\ Charles R Greathouse IV, May 23 2013

Crossrefs

Partial sums of A069905.

Sequence in context: A133523 A114805 A196722 * A000601 A062433 A317910

Adjacent sequences: A181117 A181118 A181119 * A181121 A181122 A181123

Keyword

nonn,easy

Author

Mircea Merca, Oct 04 2010

Status

approved