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A016061 a(n) = n*(n+1)*(4*n+5)/6. 27
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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]

A016061(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and n<w+x+y<=2n.  A016061(n)+A000292(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, S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.

G. Olive, Problem #504, Factorizations and Sums, Two-Year College Math. Jnl., 25 (1994), 244-245.

LINKS

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

David N. Blauch, Crystal Structure of Zinc Blende.

Q├źndrim R. Gashi, Travis Schedler, David E. Speyer, Looping of the numbers game and the alcoved hypercube, arXiv:0909.5324v1 [math.RT], Sep 29, 2009.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, see p. 233.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

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 *)

PROG

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

(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

CROSSREFS

Bisection of A002623.

Cf. A002412, A008865.

Row sums of triangle A120070.

Sequence in context: A033943 A026084 A211801 * A154154 A281868 A137976

Adjacent sequences:  A016058 A016059 A016060 * A016062 A016063 A016064

KEYWORD

nonn,easy

AUTHOR

Robert G. Wilson v

STATUS

approved

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Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)