A008383 Coordination sequence for A_4 lattice.
1, 20, 110, 340, 780, 1500, 2570, 4060, 6040, 8580, 11750, 15620, 20260, 25740, 32130, 39500, 47920, 57460, 68190, 80180, 93500, 108220, 124410, 142140, 161480, 182500, 205270, 229860, 256340, 284780, 315250, 347820, 382560, 419540, 458830, 500500, 544620
Offset: 0
References
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
- R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
- J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Magma
[n eq 0 select 1 else 5*n*(7*n^2+5)/3: n in [0..45]]; // G. C. Greubel, May 25 2023
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Maple
a:= n-> `if`(n=0, 1, 35/3*n^3+25/3*n): seq (a(n), n=0..50);
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Mathematica
CoefficientList[Series[(1+16x+36x^2+16x^3+x^4)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Dec 01 2013 *) Join[{1}, LinearRecurrence[{4, -6, 4, -1}, {20, 110, 340, 780}, 40]] (* Jean-François Alcover, Jan 07 2019 *) -
SageMath
[5*n*(7*n^2+5)/3+int(n==0) for n in range(46)] # G. C. Greubel, May 25 2023
Formula
a(n) = 5*n*(7*n^2 + 5)/3, a(0) = 1.
G.f.: (1+16*x+36*x^2+16*x^3+x^4)/(1-x)^4 = 1+10*x*(2+3*x+2*x^2)/(x-1)^4. - Colin Barker, Apr 13 2012
E.g.f.: (1/3)*(3 + 5*x*(12 + 21*x + 7*x^2)*exp(x)). - G. C. Greubel, May 25 2023