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Search: id:A035607
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| A035607 |
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Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d>=1, m >= 0). |
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13
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| 1, 1, 2, 1, 4, 2, 1, 6, 8, 2, 1, 8, 18, 12, 2, 1, 10, 32, 38, 16, 2, 1, 12, 50, 88, 66, 20, 2, 1, 14, 72, 170, 192, 102, 24, 2, 1, 16, 98, 292, 450, 360, 146, 28, 2, 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2, 1, 20, 162, 688, 1666, 2364, 1970, 952, 258, 36, 2, 1, 22, 200, 978, 2816
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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Table also gives coordination sequences of same lattices.
Rows sums are given by A001333. Rising and falling diagonals are the tribonacci numbers A000213, A001590. - Paul Barry (pbarry(AT)wit.ie), Feb 13 2003
a(d,m) also gives the number of ways to choose m squares from a 2 X (d-1) grid so that no two squares in the selection are (horizontally or vertically) adjacent. - Jacob Siehler (siehlerj(AT)wlu.edu), May 13 2006
Mirror image of triangle A113413 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 15 2006
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REFERENCES
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Munarini, Emanuele, Combinatorial properties of the antichains of a garland. Integers, 9 (2009), 353-374.
J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44.
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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 (Abstract, pdf, ps).
Siehler, J, Adjacency-free selections from a 2xN grid (Mathematica notebook)
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FORMULA
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Formulae from Roger Cuculiere (rcuculiere(AT)free.fr), Apr 10 2006:
"The generating function G(x,y) of this double sequence is the sum of a(n,p)*x^n*y^p, n=1..infty, p=0..infty, which is G(x,y)=x*(1+y)/(1-x-y-(x*y)).
"The horizontal generating function H_n(y), which generates the rows of the table : (1, 2, 2, 2, 2,...), (1, 4, 8, 12, 16, ...), (1, 6, 18, 38, 66, ...), is the sum of a(n,p)*y^p, p=0..infty, for each fixed n. This is H_n(y)=((1+y)^n)/((1-y)^n)).
"The vertical generating function V_p(x), which generates the columns of the table : (1, 1, 1, 1, 1, ...}, (2, 4, 6, 8, 10, ...), (2, 8, 18, 32, 50, ...), is the sum of a(n,p)*x^n, n=1..infty, for each fixed p. This is V_p(x)=2*((1+x)^(p-1))/((1-x)^(p+1)) for p>=1 and V_0(x)=x/(1-x)."
G.f.: (1+x)/(1-x-x*y-x^2*y). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 02 2002 (But see previous lines!)
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MAPLE
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f := proc(d, m) local i; sum( 2^i*binomial(d, i)*binomial(m-1, i-1), i=1..min(d, m)); end; # n=dimension, m=norm
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CROSSREFS
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Cf. A008288, which has g.f. 1/(1-x-x*y-x^2*y).
Adjacent sequences: A035604 A035605 A035606 this_sequence A035608 A035609 A035610
Sequence in context: A135837 A027144 A158303 this_sequence A059370 A084534 A165899
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David W. Wilson (davidwwilson(AT)comcast.net).
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