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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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16
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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
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table;
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history;
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OFFSET
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0,3
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COMMENTS
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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, 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 A. Siehler, May 13 2006
Mirror image of triangle A113413 . - Philippe DELEHAM, Oct 15 2006
The Ca1 sums lead to A126116 and the Ca2 sums lead to A070550, see A180662 for the definitions of these triangle sums. [Johannes W. Meijer, Aug 05 2011]
A035607 is jointly generated with the Delannoy triangle A008288 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=2x*u(n-1,x)+v(n-1,x). See the Mathematica section. [From Clark Kimberling, Mar 05 2012]
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REFERENCES
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M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
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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Table of n, a(n) for n=0..70.
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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From Johannes W. Meijer, Aug 05 2011: (Start)
f(d,m) = sum(binomial(floor((d-1+j)/2), d-m-1)*binomial(d-m-1, floor((d-1-j)/2)), j=0..d-1), d>=1 and 0<=m<=d-1.
f(d,m) = f(d-1,m-1) + f(d-1,m) + f(d-2,m-1) (d>=3 and 1 <= m <= d-1) with f(d,0) = 1 (d>=1) and f(d,d-1) = 2 (d>=2) (End)
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, Apr 02 2002 (But see previous lines!)
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MAPLE
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A035607 := proc(d, m) local j: add(binomial(floor((d-1+j)/2), d-m-1)*binomial(d-m-1, floor((d-1-j)/2)), j=0..d-1) end: seq(seq(A035607(d, m), m=0..d-1), d=1..11); # d=dimension, m=norm [Johannes W. Meijer, Aug 05 2011]
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A008288 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A035607 *)
(* Clark Kimberling, Mar 09 2012 *)
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CROSSREFS
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Cf. A008288, which has g.f. 1/(1-x-x*y-x^2*y).
Sequence in context: A135837 A027144 A158303 * A059370 A084534 A165899
Adjacent sequences: A035604 A035605 A035606 * A035608 A035609 A035610
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KEYWORD
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nonn,easy,tabl,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from David W. Wilson.
Maple program corrected and information added by Johannes W. Meijer, Aug 05 2011
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STATUS
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approved
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