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
Stéphane Rézel, Table of n, a(n) for n = 0..1000
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.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).
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
KEYWORD
nonn
AUTHOR
Stéphane Rézel, Nov 14 2019
STATUS
approved