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A001848
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Crystal ball sequence for 6-dimensional cubic lattice.
(Formerly M4904 N2102)
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8
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1, 13, 85, 377, 1289, 3653, 8989, 19825, 40081, 75517, 134245, 227305, 369305, 579125, 880685, 1303777, 1884961, 2668525, 3707509, 5064793, 6814249, 9041957, 11847485, 15345233, 19665841, 24957661, 31388293, 39146185, 48442297, 59511829, 72616013, 88043969
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OFFSET
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0,2
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COMMENTS
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Number of nodes of degree 12 in virtual, optimal chordal graphs of diameter d(G)=n. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
Equals binomial transform of [1, 12, 60, 160, 240, 192, 64, 0, 0, 0, ...] where (1, 12, 60, 160, 240, 192, 64) = row 6 of the Chebyshev triangle A013609. - Gary W. Adamson, Jul 19 2008
a(n) is the number of points in Z^6 that are L1 (Manhattan) distance <= n from any given point. Equivalently, due to a symmetry that is easier to see in the Delannoy numbers array (A008288), as a special case of Dmitry Zaitsev's Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= 6 from any given point. - Shel Kaphan, Jan 02 2023
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 231.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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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+x)^6 /(1-x)^7.
a(n) = (4/45)*n^6 + (4/15)*n^5 + (14/9)*n^4 + (8/3)*n^3 + (196/45)*n^2 + (46/15)*n + 1. - S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Nov 25 2002
a(n) = Sum_{k=0..min(6,n)} 2^k * binomial(6,k)* binomial(n,k). See Bump et al. - Tom Copeland, Sep 05 2014
Sum_{n >= 1} (-1)^(n+1)/(n*a(n-1)*a(n)) = log(2) - 37/60 = log(2) - (1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6). - Peter Bala, Mar 23 2024
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MAPLE
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for n from 1 to k do eval(4/45*n^6+4/15*n^5+14/9*n^4+8/3*n^3+196/45*n^2+46/15*n+1); od;
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MATHEMATICA
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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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