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%I #32 Sep 06 2023 01:15:51
%S 4,8,6,12,18,8,16,38,32,10,20,66,88,50,12,24,102,192,170,72,14,28,146,
%T 360,450,292,98,16,32,198,608,1002,912,462,128,18,36,258,952,1970,
%U 2364,1666,688,162,20,40,326,1408,3530,5336,4942,2816,978,200,22
%N Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).
%H Alois P. Heinz, <a href="/A119800/b119800.txt">Antidiagonals n = 1..141, flattened</a>
%H Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
%H 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://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H Joan Serra-Sagrista, <a href="http://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
%F A(m,n) = A(m,n-1) + A(m-1,n) + A(m-1,n-1), A(m,0)=1, A(0,0)=1, A(0,n)=2.
%e The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron.
%e The third column of the table is: 12, 38, 88, 170, 292, 462, 688, 978, 1340, ... = A035597 = number of points of L1 norm 3 in cubic lattice Z^n.
%e The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
%e The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
%e The main diagonal seems to be A050146.
%p A:= proc(m, n) option remember;
%p `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
%p end:
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..10); # _Alois P. Heinz_, Apr 21 2012
%t A[m_, n_] := A[m, n] = If[n == 0, 1, If[m == 0, 2, A[m, n-1] + A[m-1, n] + A[m-1, n-1]]]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* _Jean-François Alcover_, Mar 09 2015, after _Alois P. Heinz_ *)
%o Excel cell formula: =Z(-1)S(-1)+Z(-1)S+ZS(-1). The very first row (not included into the table) contains the initialization values: 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... The very first column (not included into the table) contains the initialization values: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... Note that the first cell is common to both the initialization row and initialization column and it equal to 1.
%Y Other versions: A035607, A113413, A122542, A266213.
%Y Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.
%K easy,nonn,tabl
%O 1,1
%A _Thomas Wieder_, Jul 30 2006, Aug 06 2006
%E Offset and typos corrected by _Alois P. Heinz_, Apr 21 2012
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