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A016061
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a(n) = n*(n+1)*(4*n+5)/6.
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30
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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
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OFFSET
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0,2
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COMMENTS
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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 n<w+x+y<=2n. a(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
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)
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REFERENCES
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P. Jena and S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.
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LINKS
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T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, see p. 233.
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FORMULA
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a(n) = Sum_{i=0..n-1} 2*i^2 + i. - Jani Nurminen (slinky(AT)iki.fi), May 14 2006
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
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MAPLE
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n*(n+1)*(4*n+5)/6 ;
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MATHEMATICA
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CoefficientList[Series[x (3 + x) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2013 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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