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Search: id:A123518
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| A123518 |
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Number of dumbbells in all possible arrangements of dumbbells on a 2 x n rectangular array of compartments. |
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+0
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| 1, 8, 38, 166, 671, 2602, 9792, 36068, 130697, 467556, 1655406, 5811290, 20255279, 70172502, 241839184, 829685064, 2835099649, 9653650752, 32768012102, 110913651342, 374469646511, 1261386990850, 4240037471152, 14225209349036
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OFFSET
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1,2
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COMMENT
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a(n)=Sum(k*A046741(n,k), k=0..n).
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REFERENCES
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R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
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FORMULA
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G.f.=z(1+2z-3z^2+2z^3)/(1-3z-z^2+z^3)^2.
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EXAMPLE
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a(2)=8 because in a 2 x 2 array of compartments, numbered clockwise starting from the NW one, we have 7 (=A030186(2)) possible arrangements of dumbbells:
[ ], [14], [23], [12], [34], [14,23], and [12,34] (ij indicates a dumbbell placed in the compartments i and j); these contain altogether 8 dumbbells.
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MAPLE
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G:=z*(1+2*z-3*z^2+2*z^3)/(1-3*z-z^2+z^3)^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=1..27);
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CROSSREFS
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Cf. A030186, A046741.
Adjacent sequences: A123515 A123516 A123517 this_sequence A123519 A123520 A123521
Sequence in context: A036684 A026640 A122682 this_sequence A007786 A026662 A003353
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 16 2006
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