OFFSET
1,1
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.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
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.
Square array A(n,k) begins:
4, 8, 12, 16, 20, 24, 28, 32, 36, ...
6, 18, 38, 66, 102, 146, 198, 258, 326, ...
8, 32, 88, 192, 360, 608, 952, 1408, 1992, ...
10, 50, 170, 450, 1002, 1970, 3530, 5890, 9290, ...
12, 72, 292, 912, 2364, 5336, 10836, 20256, 35436, ...
14, 98, 462, 1666, 4942, 12642, 28814, 59906, 115598, ...
16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, ...
18, 162, 978, 4482, 16722, 53154, 148626, 374274, 864146, ...
20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, ...
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
KEYWORD
AUTHOR
Thomas Wieder, Jul 30 2006, Aug 06 2006
EXTENSIONS
Offset and typos corrected by Alois P. Heinz, Apr 21 2012
STATUS
approved