|
%I #123 Nov 09 2023 12:14:13
%S 0,3,13,34,70,125,203,308,444,615,825,1078,1378,1729,2135,2600,3128,
%T 3723,4389,5130,5950,6853,7843,8924,10100,11375,12753,14238,15834,
%U 17545,19375,21328,23408,25619,27965,30450,33078,35853,38779,41860
%N a(n) = n*(n+1)*(4*n+5)/6.
%C Number of ZnS molecules in cluster of n layers in zinc blende crystal.
%C (Zinc sulfide crystallizes in two different forms: wurtzite and zinc blende, the latter is also spelled zincblende.) - _Jonathan Vos Post_, Jan 22 2013
%C 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
%C 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
%C 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
%C Degrees of the Hilbert polynomials for B_3 and C_3, per p. 13 of Gashi et al. - _Jonathan Vos Post_, Dec 14 2013
%C 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
%C From _J. M. Bergot_, Jun 18 2013: (Start)
%C Consider the lower half, including the main diagonal, of the array in A144216 as a triangle. The rows begin:
%C 0;
%C 1, 2;
%C 3, 4, 6;
%C 6, 7, 9, 12, ...
%C The sum of the terms in row(n) is a(n). (End)
%C 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
%D P. Jena and S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.
%H Vincenzo Librandi, <a href="/A016061/b016061.txt">Table of n, a(n) for n = 0..1000</a>
%H David N. Blauch, <a href="http://www.chm.davidson.edu/vce/crystals/ZincBlende.html">Crystal Structure of Zinc Blende</a>.
%H Qëndrim R. Gashi, Travis Schedler and David E. Speyer, <a href="http://arxiv.org/abs/0909.5324">Looping of the numbers game and the alcoved hypercube</a>, arXiv:0909.5324v1 [math.RT], 2009.
%H Johan Kok, <a href="https://pisrt.org/psr-press/journals/odam-vol-6-issue-2-2023/introduction-to-total-chromatic-vertex-stress-of-graphs/">Introduction to total chromatic vertex stress of graphs</a>, Open J. Disc. Appl. Math. (ODAM, 2023) Vol. 6, No. 2, 32-38. See p. 34.
%H T. P. Martin, <a href="https://doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, Vol. 273 (1996), pp. 199-241, see p. 233.
%H Gloria Olive, <a href="http://www.jstor.org/stable/2687658">Problem #504, Factorizations and Sums</a>, Two-Year College Math. Jnl., Vol. 25 (1994), pp. 244-245.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x*(3+x)/(1-x)^4. - _Paul Barry_, Feb 27 2003
%F Partial sums of A014105. - _Jon Perry_, Jul 23 2003
%F a(n) = Sum_{i=0..n-1} 2*i^2 + i. - Jani Nurminen (slinky(AT)iki.fi), May 14 2006
%F a(n) = 2*n^3/3 +3*n^2/2 + 5*n/6. - _Jonathan Vos Post_, Dec 14 2013
%F a(n) = (4*n+5)/(2*n+1)*A000330(n). - _Alexander R. Povolotsky_, Mar 09 2013
%F a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Bobby Milazzo_, Mar 10 2013
%F Sum_{n>=1} 1/a(n) = 12*Pi/5 + 72*log(2)/5 - 426/25. - _Amiram Eldar_, Jan 04 2022
%F E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/6. - _Stefano Spezia_, Jul 31 2022
%p A016061 := proc(n)
%p n*(n+1)*(4*n+5)/6 ;
%p end proc: # _R. J. Mathar_, Sep 26 2013
%t CoefficientList[Series[x (3 + x) / (1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 25 2013 *)
%t Table[n(n+1)(4*n+5)/6, {n,0,100}] (* _Wesley Ivan Hurt_, Sep 25 2013 *)
%o (PARI) v=vector(40,i,t(i)); s=0; forstep(i=2,40,2,s+=v[i]; print1(s","))
%o (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
%Y Bisection of A002623.
%Y Cf. A000292, A000330, A002412, A008865, A014105, A144216, A159797, A180662.
%Y Row sums of triangle A120070.
%K nonn,easy
%O 0,2
%A _Robert G. Wilson v_
|
