A008534 Coordination sequence for {A_6}* lattice.
1, 14, 98, 462, 1596, 4410, 10374, 21658, 41272, 73206, 122570, 195734, 300468, 446082, 643566, 905730, 1247344, 1685278, 2238642, 2928926, 3780140, 4818954, 6074838, 7580202, 9370536, 11484550, 13964314, 16855398, 20207012, 24072146, 28507710, 33574674, 39338208, 45867822
Offset: 0
References
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
- 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 (6,-15,20,-15,6,-1).
Programs
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GAP
Concatenation([1], List([1..45], n-> 7*n*(6+5*n^2+n^4)/6 )); # G. C. Greubel, Nov 10 2019
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Magma
[1] cat [7*n*(6+5*n^2+n^4)/6: n in [1..45]]; // G. C. Greubel, Nov 10 2019
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Maple
1, seq( (7*k^5+35*k^3+42*k)/6, k=1..40);
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Mathematica
CoefficientList[Series[(x^6 +8x^5 +29x^4 +64x^3 +29x^2 +8x +1)/(x-1)^6, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 20 2013 *) Table[If[n==0,1, 7*n*(6+5*n^2+n^4)/6], {n,0,40}] (* G. C. Greubel, Nov 10 2019 *) -
PARI
vector(46, n, if(n==1,1, 7*(n-1)*(6+5*(n-1)^2+(n-1)^4)/6 ) ) \\ G. C. Greubel, Nov 10 2019
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Sage
[1]+[7*n*(6+5*n^2+n^4)/6 for n in (1..45)]; # G. C. Greubel, Nov 10 2019
Formula
G.f.: (x^6+8*x^5+29*x^4+64*x^3+29*x^2+8*x+1)/(x-1)^6. [Conway-Sloane] - Colin Barker, Sep 21 2012
a(n) = (7/6)*n*(n^2+2)*(n^2+3) for n>0, a(0)=1. - Bruno Berselli, Feb 28 2013
E.g.f.: 1 + x*(84 + 210*x + 210*x^2 + 70*x^3 + 7*x^4)*exp(x)/6. - G. C. Greubel, Nov 10 2019
Comments