A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

1, 1, 4, 1, 6, 18, 1, 8, 32, 88, 1, 10, 50, 170, 450, 1, 12, 72, 292, 912, 2364, 1, 14, 98, 462, 1666, 4942, 12642, 1, 16, 128, 688, 2816, 9424, 27008, 68464, 1, 18, 162, 978, 4482, 16722, 53154, 148626, 374274, 1, 20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, 1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898

Offset: 0

Author

R. J. Mathar, Apr 21 2021

Keywords

• nonn
• tabl
• easy

Examples

			The full array starts
     1      2      2      2      2      2      2      2      2
     1      4      8     12     16     20     24     28     32
     1      6     18     38     66    102    146    198    258
     1      8     32     88    192    360    608    952   1408
     1     10     50    170    450   1002   1970   3530   5890
     1     12     72    292    912   2364   5336  10836  20256
     1     14     98    462   1666   4942  12642  28814  59906
     1     16    128    688   2816   9424  27008  68464 157184
     1     18    162    978   4482  16722  53154 148626 374274
		

Links

• J. Schroder, Generalized Schroder Numbers and the Rotation principle, J. Int. Seq. 10 (2007) # 07.7.7, Theorem 4.2.

Crossrefs

Cf. A035607 (by antidiags), A008574 (n=1), A005899 (n=2), A008412 (n=3), A008413 (n=4), A008414 (n=5), A001105 (k=2), A035597 (k=3), A035598 (k=4).

Main diagonal gives A050146(n+1).

Programs

Maple

A343599 := proc(n,k)
    local g,x,y ;
    g := (1+y)/(1-x-y-x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:

Mathematica

T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

Formula

G.f.: (1+y)/(1-x-y-x*y).

T(n,k) = A008288(n,k) + A008288(n,k-1).