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A008415 Coordination sequence for 7-dimensional cubic lattice.

%I A008415 #48 Apr 27 2020 06:23:19
%S A008415 1,14,98,462,1666,4942,12642,28814,59906,115598,209762,361550,596610,
%T A008415 948430,1459810,2184462,3188738,4553486,6376034,8772302,11879042,
%U A008415 15856206,20889442,27192718,35011074,44623502,56345954,70534478
%N A008415 Coordination sequence for 7-dimensional cubic lattice.
%H A008415 Seiichi Manyama, <a href="/A008415/b008415.txt">Table of n, a(n) for n = 0..10000</a>
%H A008415 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H A008415 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A008415 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A008415 G.f.: ((1+x)/(1-x))^7.
%F A008415 a(n) = 4*n*n*(2*n*n + 7)*(n*n + 14)/45 + 2 - 0^n. - _George F. Johnson_, Feb 21 2013
%F A008415 a(n) = A008414(n) + A008414(n-1) + a(n-1). - _Bruce J. Nicholson_, Dec 17 2017
%F A008415 n*a(n) = 14*a(n-1) + (n-2)*a(n-2) for n > 1. - _Seiichi Manyama_, Jun 06 2018
%t A008415 CoefficientList[Series[((1+x)/(1-x))^7,{x,0,30}],x] (* _Harvey P. Dale_, Oct 11 2015 *)
%o A008415 (Python)
%o A008415 R = []
%o A008415 for n in range(29):
%o A008415     r=4*n*n*(2*n*n +7)*(n*n +14)//45 +2-0**n
%o A008415     R.append(r)
%o A008415 print(R)
%o A008415 # _George F. Johnson_ Feb 02 2013
%o A008415 (PARI) a(n) = 2*(4*n^6+70*n^4+196*n^2+45)/45-0^n; \\ _Altug Alkan_, Dec 18 2017
%Y A008415 Cf. A008414.
%K A008415 nonn,easy
%O A008415 0,2
%A A008415 _N. J. A. Sloane_