A004188 a(n) = n*(3*n^2 - 1)/2.

0, 1, 11, 39, 94, 185, 321, 511, 764, 1089, 1495, 1991, 2586, 3289, 4109, 5055, 6136, 7361, 8739, 10279, 11990, 13881, 15961, 18239, 20724, 23425, 26351, 29511, 32914, 36569, 40485, 44671, 49136, 53889, 58939, 64295, 69966, 75961

Offset: 0

Author

Albert D. Rich (Albert_Rich(AT)msn.com)

Keywords

• nonn
• easy

Comments

3-dimensional analog of centered polygonal numbers.

(1), (4+7), (10+13+16), (19+22+25+28), ... - Jon Perry, Sep 10 2004

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
• T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Cf. A002412, A016061, A051682.

Cf. A236770 (partial sums).

Programs

Magma

[n*(3*n^2-1)/2: n in [0..50]]; //Vincenzo Librandi, May 15 2011

Maple

seq(binomial(2*n+1,3)+binomial(n+1,3), n=0..37); # Zerinvary Lajos, Jan 21 2007

Mathematica

Table[n(3n^2-1)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,11,39},40] (* Harvey P. Dale, Jul 19 2019 *)

PARI

vector(40, n, n*(3*n^2-1)/2)

Formula

Partial sums of n-1 3-spaced triangular numbers, e.g., a(4) = t(1) + t(4) + t(7) = 1 + 10 + 28 = 39. - Jon Perry, Jul 23 2003

a(n) = C(2*n+1,3) + C(n+1,3), n >= 0. - Zerinvary Lajos, Jan 21 2007

a(n) = A000447(n) + A000292(n). - Zerinvary Lajos, Jan 21 2007

G.f.: x*(1+7*x+x^2) / (x-1)^4. - R. J. Mathar, Oct 08 2011

From Miquel Cerda, Dec 25 2016: (Start)

a(n) = A000578(n) + A135503(n).

a(n) = A007588(n) - A135503(n). (End)

E.g.f.: (x/2)*(2 + 9*x + 3*x^2)*exp(x). - G. C. Greubel, Sep 01 2017