OFFSET
1,7
COMMENTS
Independence number of the kings' graph on toroidal n X m chessboard.
Right border T(n,n) is A189889.
For the usual non-toroidal case, the formula is ceiling(m/2)*ceiling(n/2).
REFERENCES
John J. Watkins, Across the Board: The Mathematics of Chessboard Problem, Princeton University Press, 2004, pages 194-196.
LINKS
Indranil Ghosh, Rows 1..125, flattened
V. Kotesovec, Non-attacking chess pieces.
FORMULA
T(n,m) = floor(min(m*floor(n/2), n*floor(m/2))/2) for m>1;
T(n,1) = floor(n/2) for n>1.
EXAMPLE
Triangle starts:
1;
1, 1;
1, 1, 1;
2, 2, 2, 4;
2, 2, 2, 4, 5;
3, 3, 3, 6, 6, 9;
3, 3, 3, 6, 7, 9, 10;
...
MATHEMATICA
T[1, 1] = 1; T[n_, m_]:= If[m==1, Floor[n/2], Floor[Min[m Floor[n/2], n Floor[m/2]]/2]]; Flatten[Table[T[n, m], {n, 1, 12}, {m, 1, n}]] (* Indranil Ghosh, Mar 09 2017 *)
PROG
(PARI) tabl(nn) = {for(n=1, 12, for(m=1, n, print1(if(m==1, if(n==1, 1, floor(n/2)), floor(min(m*floor(n/2), n*floor(m/2))/2)), ", "); ); print(); ); };
tabl(12); \\ Indranil Ghosh, Mar 09 2017
(Python)
def T(n, m):
....if m==1:
........if n==1: return 1
........return n/2
....return min(m*(n/2), n*(m/2))/2
i=1
for n in range(1, 126):
....for m in range(1, n+1):
........print str(i)+" "+str(T(n, m))
........i+=1 # Indranil Ghosh, Mar 09 2017
CROSSREFS
KEYWORD
AUTHOR
Andrey Zabolotskiy, Dec 16 2016
STATUS
approved