A059963 Triangle T(n,k) giving number of ways of placing n nonattacking queens on n X n board with the queen on the first row fixed at column k, 1<=k<=n.

1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 2, 2, 2, 2, 0, 1, 1, 1, 1, 0, 4, 7, 6, 6, 6, 7, 4, 4, 8, 16, 18, 18, 16, 8, 4, 28, 30, 47, 44, 54, 44, 47, 30, 28, 64, 48, 65, 93, 92, 92, 93, 65, 48, 64, 96, 219, 209, 295, 346, 350, 346, 295, 209, 219, 96, 500, 806, 1165, 1359, 1631, 1639

Offset

1,11

Comments

A000170 (non-attacking queens) can be derived from this sequence as follows: a(12)= 2*(S1(12)+S2(12)+S3(12)+S4(12)+S5(12)+S6(12)) when n is even, a(13)=S7(13) + 2*(S1(13)+S2(13)+S3(13)+S4(13)+S5(13)+S6(13)) when n is odd. Here Si(j) means T(j,i). - Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), Jan 05 2004

Links

Table of n, a(n) for n=1..72.

Patrick R. GUILLEMIN, Extension of triangle to 22 rows

Patrick R. GUILLEMIN, Extension of triangle to 22 rows

Example

When n = 8 there are 16 ways to place if the queen on the first row is at the third column
Triangle begins:
1,
0,0,
0,0,0,
0,1,1,0,
2,2,2,2,2,
0,1,1,1,1,0,
4,7,6,6,6,7,4,
4,8,16,18,18,16,8,4,
28,30,47,44,54,44,47,30,28, etc.

Crossrefs

Cf. A000170.

Sequence in context: A037868 A173069 A363934 * A137934 A133738 A240713

Adjacent sequences: A059960 A059961 A059962 * A059964 A059965 A059966

Keyword

nonn,tabl

Author

Yong Kong (ykong(AT)curagen.com), Mar 03 2001

Extensions

Confirmed by Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), who, together with colleagues, has computed the first 21 rows of this triangle, Jan 05 2004

Sep 15 2004: Patrick R. GUILLEMIN (patrick.guillemin(AT)etsi.org), together with colleagues, has computed the 22nd row of this triangle.

Status

approved