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A008399
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Coordination sequence for E_6 lattice.
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2
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1, 72, 1062, 6696, 26316, 77688, 189810, 405720, 785304, 1408104, 2376126, 3816648, 5885028, 8767512, 12684042, 17891064, 24684336, 33401736, 44426070, 58187880, 75168252, 95901624, 120978594, 151048728, 186823368, 229078440, 278657262, 336473352, 403513236
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OFFSET
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0,2
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REFERENCES
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M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
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LINKS
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J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
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FORMULA
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a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.
Bacher et al. give a g.f.
G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^6 = 1 + 18*x*(4+35*x+78*x^2+35*x^3+4*x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/5)*x*(360 + 2295*x + 3105*x^2 + 1170*x^3 + 117*x^4 )*exp(x). - G. C. Greubel, May 29 2023
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MAPLE
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1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);
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MATHEMATICA
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LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 72, 1062, 6696, 26316, 77688, 189810}, 30] (* Harvey P. Dale, Oct 24 2022 *)
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PROG
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(Magma) [1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // G. C. Greubel, May 29 2023
(SageMath) [9*n*(13*n^2+7)*(n^2+1)//5 +int(n==0) for n in range(41)] # G. C. Greubel, May 29 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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