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A329451
Maximum number of pieces that can be captured during one move on an n X n board according to the international draughts capture rules.
1
0, 0, 0, 1, 1, 4, 5, 9, 10, 16, 19, 25, 28, 36, 41, 49, 54, 64, 71, 81, 88, 100, 109, 121, 130, 144, 155, 169, 180, 196, 209, 225, 238, 256, 271, 289, 304, 324, 341, 361, 378, 400, 419, 441, 460, 484, 505, 529, 550, 576, 599, 625, 648, 676, 701, 729, 754, 784
OFFSET
0,6
COMMENTS
Captures are made diagonally, forward and backward. Kings have the long-range capturing capability. During the multiple capture, a piece may pass over the same empty square several times, but no opposing piece can be jumped twice. Captured pieces can only be lifted from the board after the end of the multiple capture.
LINKS
Fabien Gigante, Solution to Problem J122 La grande rafle de la dame, Diophante, 2009 (in French).
Stéphane Rézel, Solution to Problem J128 La méga-rafle de la dame, Diophante, 2020 (in French).
World Draughts Federation, Official rules for international draughts.
FORMULA
a(2*t+1) = t^2 = A000290(t).
a(4*t+6) = 4*t^2 + 10*t + 5 = A125202(t+2).
a(4*t+8) = 4*t^2 + 14*t + 10 = A059193(t+2).
a(0) = a(2) = 0; a(4) = 1.
Recurrence: For t >= 1, a(2*t+1) = a(2*t-1) + 2*t - 1;
For t >= 1, a(4*t+3) = a(4*t+2) + 2*t + 2; a(4*t+2) = a(4*t+1) + 2*t - 1;
For t >= 2, a(4*t+1) = a(4*t) + 2*t + 2; a(4*t) = a(4*t-1) + 2*t - 3.
From Colin Barker, Nov 14 2019: (Start)
G.f.: x^3*(1 - x + 3*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 2*x^6 - x^7) / ((1 - x)^3*(1 + x)*(1 + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>10.
(End)
EXAMPLE
It is possible to capture in a single move 19 opposing pieces on a 10 X 10 board, but not one more, so a(10) = 19.
PROG
(PARI) a(n) = if(n<5, floor(n/3), (n^2 - 2*n + if(n%2, 1, 2*(n%4) - 8))/4)
CROSSREFS
Cf. A000290, A059193, A125202, A000982 (active squares).
Sequence in context: A059582 A257058 A189889 * A308578 A034809 A336900
KEYWORD
nonn
AUTHOR
Stéphane Rézel, Nov 14 2019
STATUS
approved