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A279409 Triangle read by rows: T(n,m) (n>=m>=1) = maximum number of nonattacking kings on an n X m toroidal board. 2
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, 4, 4, 4, 8, 8, 12, 12, 16, 4, 4, 4, 8, 9, 12, 13, 16, 18, 5, 5, 5, 10, 10, 15, 15, 20, 20, 25, 5, 5, 5, 10, 11, 15, 16, 20, 22, 25, 27, 6, 6, 6, 12, 12, 18, 18, 24, 24, 30, 30, 36 (list; table; graph; refs; listen; history; text; internal format)
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

Dan Freeman, Chessboard Puzzles Part 4 - Other Surfaces and Variations.

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

Cf. A085801, A189889, A279408.

Sequence in context: A102298 A049298 A075016 * A102445 A160691 A049716

Adjacent sequences:  A279406 A279407 A279408 * A279410 A279411 A279412

KEYWORD

nonn,tabl,easy

AUTHOR

Andrey Zabolotskiy, Dec 16 2016

STATUS

approved

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Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)