A016061 a(n) = n*(n+1)*(4*n+5)/6.

0, 3, 13, 34, 70, 125, 203, 308, 444, 615, 825, 1078, 1378, 1729, 2135, 2600, 3128, 3723, 4389, 5130, 5950, 6853, 7843, 8924, 10100, 11375, 12753, 14238, 15834, 17545, 19375, 21328, 23408, 25619, 27965, 30450, 33078, 35853, 38779, 41860

Offset: 0

Author

Robert G. Wilson v

Keywords

• nonn
• easy

Comments

Number of ZnS molecules in cluster of n layers in zinc blende crystal.

(Zinc sulfide crystallizes in two different forms: wurtzite and zinc blende, the latter is also spelled zincblende.) - Jonathan Vos Post, Jan 22 2013

The Kn4 triangle sums of the Connell-Pol triangle A159797 lead to the sequence given above. For the definitions of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, May 20 2011

If one generated primitive Pythagorean triangles (2n+1, 2n+3) the collective sum of their perimeters for each n is four times the numbers listed in this sequence. - J. M. Bergot, Jul 18 2011

a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and nA000292(n)+A000292(n+1)=n^3. - Clark Kimberling, Jun 04 2012

Degrees of the Hilbert polynomials for B_3 and C_3, per p. 13 of Gashi et al. - Jonathan Vos Post, Dec 14 2013

Number of solutions to a + b = c + d when 0 < a <= k, 0 <= b, c, d <= k, k = 0, 1, 2, 3.... Taken from Step 1 2007 problem #1(i) using 4 digit balanced numbers. - Bobby Milazzo, Mar 09 2013

From J. M. Bergot, Jun 18 2013: (Start)

Consider the lower half, including the main diagonal, of the array in A144216 as a triangle. The rows begin:

0;

1, 2;

3, 4, 6;

6, 7, 9, 12, ...

The sum of the terms in row(n) is a(n). (End)

This sequence is related to A008865 by a(n) = n*A008865(n+1) - Sum_{i=1..n} A008865(i) for n>0. - Bruno Berselli, Aug 06 2015

References

• P. Jena and S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• David N. Blauch, Crystal Structure of Zinc Blende.
• Qëndrim R. Gashi, Travis Schedler and David E. Speyer, Looping of the numbers game and the alcoved hypercube, arXiv:0909.5324v1 [math.RT], 2009.
• Johan Kok, Introduction to total chromatic vertex stress of graphs, Open J. Disc. Appl. Math. (ODAM, 2023) Vol. 6, No. 2, 32-38. See p. 34.
• T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, see p. 233.
• Gloria Olive, Problem #504, Factorizations and Sums, Two-Year College Math. Jnl., Vol. 25 (1994), pp. 244-245.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Bisection of A002623.

Cf. A000292, A000330, A002412, A008865, A014105, A144216, A159797, A180662.

Row sums of triangle A120070.

Programs

Magma

I:=[0,3,13,34]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 25 2013

Maple

A016061 := proc(n)
    n*(n+1)*(4*n+5)/6 ;
end proc: # R. J. Mathar, Sep 26 2013

Mathematica

CoefficientList[Series[x (3 + x) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2013 *)
Table[n(n+1)(4*n+5)/6, {n,0,100}] (* Wesley Ivan Hurt, Sep 25 2013 *)

PARI

v=vector(40,i,t(i)); s=0; forstep(i=2,40,2,s+=v[i]; print1(s","))

Formula

G.f.: x*(3+x)/(1-x)^4. - Paul Barry, Feb 27 2003

Partial sums of A014105. - Jon Perry, Jul 23 2003

a(n) = Sum_{i=0..n-1} 2*i^2 + i. - Jani Nurminen (slinky(AT)iki.fi), May 14 2006

a(n) = 2*n^3/3 +3*n^2/2 + 5*n/6. - Jonathan Vos Post, Dec 14 2013

a(n) = (4*n+5)/(2*n+1)*A000330(n). - Alexander R. Povolotsky, Mar 09 2013

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Bobby Milazzo, Mar 10 2013

Sum_{n>=1} 1/a(n) = 12*Pi/5 + 72*log(2)/5 - 426/25. - Amiram Eldar, Jan 04 2022

E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/6. - Stefano Spezia, Jul 31 2022