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A380577 a(n) is the number of distinct compositions of chess pieces with a collective material value of n that one color in a game can have, where 0 <= n <= 103.

%I A380577 #9 Apr 04 2025 17:19:15
%S A380577 1,1,1,3,3,4,7,7,9,13,14,17,22,24,28,35,38,41,52,54,59,72,73,79,95,95,
%T A380577 101,117,120,122,144,139,146,166,159,165,186,174,184,195,189,199,204,
%U A380577 197,201,208,204,194,206,194,193,195,182,182,178,177,159,177,142,154,137,145,122,135,103,121,96,104,85,96,71,77,63,73,52,60,45,48,40,41,31,39,23,26,23,22,18,18,11,15,8,10,9,6,5,4,2,5,1,1,2,0,0,1
%N A380577 a(n) is the number of distinct compositions of chess pieces with a collective material value of n that one color in a game can have, where 0 <= n <= 103.
%C A380577 The following (standard) values of the chess pieces are used here: pawn (P): 1, bishop (B): 3, knight (K): 3, rook (R): 5, queen (Q): 9. The King is always considered to be included and has the value 0.
%C A380577 The game begins with the piece numbers P = 8, B = 2, K = 2, R = 2 and Q = 1 and with the collective material value of 1*8 + 3*2 + 3*2 + 5*2 + 9*1 = 39. Pieces can be lost and pawns can be converted into one of the other four piece types. Within these rules (see also ranges and inequalities in the Maple program), a(n) is the number of nonnegative integer solutions to 1*P + 3*B + 3*K + 5*R + 9*Q = n.
%C A380577 The smallest collective material value of a chess piece composition is 0 (king alone), the largest 1*0 + 3*2 + 3*2 + 5*2 + 9*9 = 103 (all pawns converted into queens, no piece lost). Therefore, the definition range for n is restricted to 0 <= n <= 103 and this sequence is finite by definition.
%C A380577 There are a total of Sum_{n, n=0..103} a(n) = 8694 distinct compositions of chess pieces that one color in a game can have.
%H A380577 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chess_piece_relative_value">Chess Piece Relative Value</a>
%F A380577 a(n) <= A378248(n).
%e A380577 a(5) = 4 because exactly 4 possible chess piece compositions (P, B, K, R, Q) satisfy 1*P + 3*B + 3*K + 5*R + 9*Q = 5: (0, 0, 0, 1, 0), (2, 0, 1, 0, 0), (2, 1, 0, 0, 0), (5, 0, 0, 0, 0).
%e A380577 a(97) = 5 because exactly 5 possible chess piece compositions (P, B, K, R, Q) satisfy 1*P + 3*B + 3*K + 5*R + 9*Q = 97: (0, 0, 2, 2, 9), (0, 1, 1, 2, 9), (0, 2, 0, 1, 9), (0, 2, 3, 2, 8), (0, 3, 2, 2, 8).
%p A380577 A380577:=proc(n)
%p A380577     local P,B,K,R,Q,a;
%p A380577     a:=0;
%p A380577     for P from 0 to 8 do
%p A380577         for B from 0 to 10 do
%p A380577           for K from 0 to 10 do
%p A380577 	          for R from 0 to 10 do
%p A380577 	            for Q from 0 to 9 do
%p A380577 	              if P+3*B+3*K+5*R+9*Q=n and P+B<=10 and P+K<=10 and P+R<=10 and P+Q<=9 and P+B+K<=12 and P+B+R<=12 and P+B+Q<=11 and P+K+R<=12 and P+K+Q<=11 and P+R+Q<=11 and P+B+K+R<=14 and P+B+K+Q<=13 and P+B+R+Q<=13 and P+K+R+Q<=13 and P+B+K+R+Q<=15 then
%p A380577 	                 a:=a+1
%p A380577 	              fi
%p A380577 	            od
%p A380577 	          od
%p A380577 	        od
%p A380577        od
%p A380577     od;
%p A380577     return a
%p A380577 end proc;
%p A380577 seq(A380577(n),n=0..103);
%Y A380577 Cf. A278832, A378248.
%K A380577 nonn,fini,full
%O A380577 0,4
%A A380577 _Felix Huber_, Mar 30 2025