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A008402
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Crystal ball sequence for {E_6}* lattice.
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2
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1, 55, 883, 6085, 26461, 86491, 232975, 545833, 1151065, 2235871, 4065931, 7004845, 11535733, 18284995, 28048231, 41818321, 60815665, 86520583, 120707875, 165483541, 223323661, 297115435
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history;
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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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G.f.: (1 + 48*x + 519*x^2 + 1024*x^3 + 519*x^4 + 48*x^5 + x^6)/(1-x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=1, a(1)=55, a(2)=883, a(3)=6085, a(4)=26461, a(5)=86491, a(6)=232975. - Harvey P. Dale, Jun 20 2013
a(n) = 3*n^6 + 9*n^5 + 15*n^4 + 15*n^3 + 9*n^2 + 3*n + 1 = 1 + 3*n*(n+1)*(n^2+n+1)^2. - Charles R Greathouse IV, Jun 20 2013
E.g.f.: exp(x)*(1 + 54*x + 387*x^2 + 600*x^3 + 300*x^4 + 54*x^5 + 3*x^6). - Stefano Spezia, Apr 15 2022
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MATHEMATICA
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CoefficientList[Series[(1+48x+519x^2+1024x^3+519x^4+48x^5+x^6)/(1-x)^7, {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 55, 883, 6085, 26461, 86491, 232975}, 30] (* Harvey P. Dale, Jun 20 2013 *)
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PROG
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(Magma) [1 +3*n*(n+1)*(n^2+n+1)^2: n in [0..40]]; // G. C. Greubel, May 31 2023
(SageMath) [1 +3*n*(n+1)*(n^2+n+1)^2 for n in range(41)] # G. C. Greubel, May 31 2023
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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