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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.

%I A329451 #36 Aug 22 2021 00:16:22
%S A329451 0,0,0,1,1,4,5,9,10,16,19,25,28,36,41,49,54,64,71,81,88,100,109,121,
%T A329451 130,144,155,169,180,196,209,225,238,256,271,289,304,324,341,361,378,
%U A329451 400,419,441,460,484,505,529,550,576,599,625,648,676,701,729,754,784
%N 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.
%C A329451 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.
%H A329451 Stéphane Rézel, <a href="/A329451/b329451.txt">Table of n, a(n) for n = 0..1000</a>
%H A329451 Fabien Gigante, <a href="http://www.diophante.fr/images/stories/diophante/rubriques_J/J122FG.pdf">Solution</a> to Problem <a href="http://www.diophante.fr/problemes-par-themes/jeux-de-plateaux/46-j-jeux-de-plateaux/1218-j122-la-grande-rafle-de-la-dame">J122 La grande rafle de la dame</a>, Diophante, 2009 (in French).
%H A329451 Stéphane Rézel, <a href="http://www.diophante.fr/images/stories/diophante/janvier_2020-03/J128SR.pdf">Solution</a> to Problem <a href="http://www.diophante.fr/problemes-par-themes/jeux-de-plateaux/4604-j128-la-mega-rafle-de-la-dame">J128 La méga-rafle de la dame</a>, Diophante, 2020 (in French).
%H A329451 World Draughts Federation, <a href="http://www.fmjd.org/downloads/FMJD_Annexes_2018.pdf">Official rules for international draughts</a>.
%H A329451 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A329451 a(2*t+1) = t^2 = A000290(t).
%F A329451 a(4*t+6) = 4*t^2 + 10*t + 5 = A125202(t+2).
%F A329451 a(4*t+8) = 4*t^2 + 14*t + 10 = A059193(t+2).
%F A329451 a(0) = a(2) = 0; a(4) = 1.
%F A329451 Recurrence: For t >= 1, a(2*t+1) = a(2*t-1) + 2*t - 1;
%F A329451 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;
%F A329451 For t >= 2, a(4*t+1) = a(4*t) + 2*t + 2; a(4*t) = a(4*t-1) + 2*t - 3.
%F A329451 From _Colin Barker_, Nov 14 2019: (Start)
%F A329451 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)).
%F A329451 a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>10.
%F A329451 (End)
%e A329451 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.
%o A329451 (PARI) a(n) = if(n<5, floor(n/3), (n^2 - 2*n + if(n%2, 1, 2*(n%4) - 8))/4)
%Y A329451 Cf. A000290, A059193, A125202, A000982 (active squares).
%K A329451 nonn
%O A329451 0,6
%A A329451 _Stéphane Rézel_, Nov 14 2019