A378248 Number of sets of chess pieces whose collective material value adds to n.

1, 1, 1, 3, 3, 4, 7, 7, 9, 14, 15, 18, 25, 27, 32, 42, 45, 52, 66, 71, 81, 99, 106, 120, 143, 153, 171, 200, 214, 237, 273, 291, 320, 364, 387, 423, 476, 505, 549, 612, 648, 701, 775, 819, 882, 969, 1022, 1096, 1197, 1260, 1347, 1463, 1537, 1638, 1771, 1858

Offset

0,4

Comments

The pieces are valued pawn=1, knight=3, bishop=3, rook=5, queen=9.

The knight and bishop are different ways to add 3 into the total.

So a(n) is the number of ways to partition n into a sum of parts 1, 3, 5, 9 with two types of 3.

Links

Table of n, a(n) for n=0..55.

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,1,-2,1,-2,3,-1,1,-3,2,-1,2,-1,2,-2,0,-1,1).

Formula

G.f.: 1/((1-x)*(1-x^3)^2*(1-x^5)*(1-x^9)). - Andrew Howroyd, Nov 20 2024

Example

For n=4, the a(4)=3 sets of pieces are {1, 1, 1, 1}, {3a, 1}, {3b, 1}, where the knight and bishop both worth 3 are labeled 3a and 3b.
For n=9 the a(9)=14 solutions are {1, 1, 1, 1, 1, 1, 1, 1, 1}, {3a, 1, 1, 1, 1, 1, 1}, {3b, 1, 1, 1, 1, 1, 1}, {3a, 3a, 1, 1, 1}, {3a, 3b, 1, 1, 1}, {3b, 3b, 1, 1, 1}, {3a, 3a, 3a}, {3a, 3a, 3b}, {3a, 3b, 3b}, {3b, 3b, 3b}, {5, 1, 1, 1, 1}, {5, 3a, 1}, {5, 3b, 1}, and {9}.

Crossrefs

Cf. A029041.

Sequence in context: A007448 A155689 A051263 * A058674 A349252 A152651

Adjacent sequences: A378245 A378246 A378247 * A378249 A378250 A378251

Keyword

nonn,easy

Author

David W. Ziegler, Nov 20 2024

Status

approved