Search: id:A035607
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%I A035607
%S A035607 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,
%T A035607 72,170,192,102,24,2,1,16,98,292,450,360,146,28,2,1,18,128,462,912,1002,
%U A035607 608,198,32,2,1,20,162,688,1666,2364,1970,952,258,36,2,1,22,200,978,2816
%N A035607 Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read 
               by antidiagonals (d>=1, m >= 0).
%C A035607 Table also gives coordination sequences of same lattices.
%C A035607 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
%C A035607 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
%C A035607 Mirror image of triangle A113413 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 15 2006
%D A035607 Munarini, Emanuele, Combinatorial properties of the antichains of a garland. 
               Integers, 9 (2009), 353-374.
%D A035607 J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information 
               Processing Letters, 76, no. 1-2 (2000), 39-44.
%H A035607 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination 
               Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http:/
               /www.research.att.com/~njas/doc/ldl7.txt">Abstract</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.pdf">pdf</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.ps">ps</a>).
%H A035607 Siehler, J, <a href="http://home.wlu.edu/~siehlerj/computing/index.html#af">
               Adjacency-free selections from a 2xN grid</a> (Mathematica notebook)
%F A035607 Formulae from Roger Cuculiere (rcuculiere(AT)free.fr), Apr 10 2006:
%F A035607 "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)).
%F A035607 "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)).
%F A035607 "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)."
%F A035607 G.f.: (1+x)/(1-x-x*y-x^2*y). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 02 2002 (But see previous lines!)
%p A035607 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
%Y A035607 Cf. A008288, which has g.f. 1/(1-x-x*y-x^2*y).
%Y A035607 Sequence in context: A135837 A027144 A158303 this_sequence A059370 A084534 
               A165899
%Y A035607 Adjacent sequences: A035604 A035605 A035606 this_sequence A035608 A035609 
               A035610
%K A035607 nonn,easy,tabl
%O A035607 0,3
%A A035607 N. J. A. Sloane (njas(AT)research.att.com).
%E A035607 More terms from David W. Wilson (davidwwilson(AT)comcast.net).

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