%I #37 Sep 06 2023 01:41:55

%S 1,112,2592,25424,149568,629808,2100832,5910288,14610560,32641008,

%T 67232416,129565392,236214464,410909616,686647008,1108180624,

%U 1734926592,2644311920,3935599392,5734220368

%N Coordination sequence for D_8 lattice.

%H T. D. Noe, <a href="/A008361/b008361.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256

%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://dx.doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

%H 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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = 16/315*n*(219 + 1036*n^2 + 826*n^4 + 124*n^6). - _Harvey P. Dale_, Feb 21 2012

%F a(0)=1, a(1)=112, a(2)=2592, a(3)=25424, a(4)=149568, a(5)=629808, a(6)=2100832, a(7)=5910288, a(8)=14610560, a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - _Harvey P. Dale_, Feb 21 2012

%F G.f.: (x^8 + 104*x^7 + 1724*x^6 + 7768*x^5 + 12550*x^4 + 7768*x^3 + 1724*x^2 + 104*x + 1)/(x - 1)^8. - _Colin Barker_, Sep 26 2012

%p 1984/315*n^7+1888/45*n^5+2368/45*n^3+1168/105*n;

%t Join[{1},Table[16/315*n*(219+1036*n^2+826*n^4+124*n^6),{n,30}]] (* or *) Join[{1},LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{112,2592,25424,149568,629808,2100832,5910288,14610560},30]] (* _Harvey P. Dale_, Feb 21 2012 *)

%o (PARI) a(n) = 16/315*n*(219 + 1036*n^2 + 826*n^4 + 124*n^6) \\ _Charles R Greathouse IV_, Feb 10 2017

%Y A row of array A103903.

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_ and _J. H. Conway_