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A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).

%I A119800 #35 Oct 18 2024 17:55:43
%S A119800 4,8,6,12,18,8,16,38,32,10,20,66,88,50,12,24,102,192,170,72,14,28,146,
%T A119800 360,450,292,98,16,32,198,608,1002,912,462,128,18,36,258,952,1970,
%U A119800 2364,1666,688,162,20,40,326,1408,3530,5336,4942,2816,978,200,22
%N A119800 Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).
%H A119800 Alois P. Heinz, <a href="/A119800/b119800.txt">Antidiagonals n = 1..141, flattened</a>
%H A119800 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 A119800 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 A119800 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 A119800 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 A119800 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 A119800 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 A119800 The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420.
%e A119800 The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603.
%e A119800 The main diagonal seems to be A050146.
%e A119800 Square array A(n,k) begins:
%e A119800    4,   8,   12,   16,    20,    24,     28,     32,      36, ...
%e A119800    6,  18,   38,   66,   102,   146,    198,    258,     326, ...
%e A119800    8,  32,   88,  192,   360,   608,    952,   1408,    1992, ...
%e A119800   10,  50,  170,  450,  1002,  1970,   3530,   5890,    9290, ...
%e A119800   12,  72,  292,  912,  2364,  5336,  10836,  20256,   35436, ...
%e A119800   14,  98,  462, 1666,  4942, 12642,  28814,  59906,  115598, ...
%e A119800   16, 128,  688, 2816,  9424, 27008,  68464, 157184,  332688, ...
%e A119800   18, 162,  978, 4482, 16722, 53154, 148626, 374274,  864146, ...
%e A119800   20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...
%p A119800 A:= proc(m, n)  option remember;
%p A119800       `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1)))
%p A119800     end:
%p A119800 seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # _Alois P. Heinz_, Apr 21 2012
%t A119800 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 A119800 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 A119800 Other versions: A035607, A113413, A122542, A266213.
%Y A119800 Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146.
%K A119800 easy,nonn,tabl
%O A119800 1,1
%A A119800 _Thomas Wieder_, Jul 30 2006, Aug 06 2006
%E A119800 Offset and typos corrected by _Alois P. Heinz_, Apr 21 2012