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.

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, 101, 117, 120, 122, 144, 139, 146, 166, 159, 165, 186, 174, 184, 195, 189, 199, 204, 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

Offset: 0

Author

Felix Huber, Mar 30 2025

Keywords

• nonn
• fini
• full

Comments

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.

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.

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.

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.

Examples

			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).
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).
		

Links

• Wikipedia, Chess Piece Relative Value

Crossrefs

Cf. A278832, A378248.

Programs

Maple

A380577:=proc(n)
    local P,B,K,R,Q,a;
    a:=0;
    for P from 0 to 8 do
        for B from 0 to 10 do
          for K from 0 to 10 do
	          for R from 0 to 10 do
	            for Q from 0 to 9 do
	              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
	                 a:=a+1
	              fi
	            od
	          od
	        od
       od
    od;
    return a
end proc;
seq(A380577(n),n=0..103);

Formula

a(n) <= A378248(n).