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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). 15
 4, 8, 6, 12, 18, 8, 16, 38, 32, 10, 20, 66, 88, 50, 12, 24, 102, 192, 170, 72, 14, 28, 146, 360, 450, 292, 98, 16, 32, 198, 608, 1002, 912, 462, 128, 18, 36, 258, 952, 1970, 2364, 1666, 688, 162, 20, 40, 326, 1408, 3530, 5336, 4942, 2816, 978, 200, 22 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 REFERENCES J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389. J. Serra-Sagrista, Enumeration of lattice points in l_1 norm, Information Processing Letters, 76, no. 1-2 (2000), 39-44. LINKS Alois P. Heinz, Antidiagonals n = 1..141, flattened Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3. FORMULA 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. EXAMPLE The second row of the table is: 6, 18, 38, 66, 102, 146, 198, 258, 326, ... = A005899 = number of points on surface of octahedron. 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. The first rows are: A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420. The first columns are: A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603. The main diagonal seems to be A050146. MAPLE A:= proc(m, n) option remember; `if`(n=0, 1, `if`(m=0, 2, A(m, n-1) +A(m-1, n) +A(m-1, n-1))) end: seq(seq(A(n, 1+d-n), n=1..d), d=1..10); # Alois P. Heinz, Apr 21 2012 MATHEMATICA 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 *) PROG 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. CROSSREFS Other versions: A035607, A113413, A122542, A266213. Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008416, A008418, A008420, A005843, A005843, A001105, A035597, A035598, A035599, A035600, A035601, A035602, A035603, A050146. Sequence in context: A288189 A335159 A064494 * A330685 A063723 A323057 Adjacent sequences: A119797 A119798 A119799 * A119801 A119802 A119803 KEYWORD easy,nonn,tabl AUTHOR Thomas Wieder, Jul 30 2006, Aug 06 2006 EXTENSIONS Offset and typos corrected by Alois P. Heinz, Apr 21 2012 STATUS approved

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Last modified June 2 15:21 EDT 2023. Contains 363098 sequences. (Running on oeis4.)