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A141843 Triangular array T(n,k) (n >= 1, 1 <= k <= n) read by rows: row n gives the lexicographically earliest solution to the n queens problem, or n zeros if no solution exists. The k-th queen is placed in square (k, T(n, k)). 5
1, 0, 0, 0, 0, 0, 2, 4, 1, 3, 1, 3, 5, 2, 4, 2, 4, 6, 1, 3, 5, 1, 3, 5, 7, 2, 4, 6, 1, 5, 8, 6, 3, 7, 2, 4, 1, 3, 6, 8, 2, 4, 9, 7, 5, 1, 3, 6, 8, 10, 5, 9, 2, 4, 7, 1, 3, 5, 7, 9, 11, 2, 4, 6, 8, 10, 1, 3, 5, 8, 10, 12, 6, 11, 2, 7, 9, 4, 1, 3, 5, 2, 9, 12, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

The sequence now contains the solutions up to n=61.

History: In December, work of Matteo Fischetti and Domenico Salvagnin, using Integer Linear Programming (ILP), found solutions for n=56 to n=61; they also found solutions for higher n, but not in contiguous sequence.

Solutions for n=48 to n=55 were found by Wolfram Schubert, around 2010, but not entered in the OEIS.

Entries for board size 46 X 46 (a new solution) and for board size 47 X 47 (already known to Colin Pearson) were added to this sequence in November 2011.

The solution for the 46 X 46 board was discovered by Matthias Engelhardt on Apr 30 2011, the solution for the 47 X 47 board by Colin S. Pearson on Jan 09 2008.

The main contributor for this sequence can be contacted via a 'feedback' form at http://queens.cspea.co.uk/

LINKS

Matthias Engelhardt, Table of n, a(n) for n = 1..1892, (previous version of Colin Pearson enlarged)

Matthias R. Engelhardt, The old nQueens problem

Matteo Fischetti, Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.

Colin S. Pearson, CSP Queens - Counting Queen-placements

Martin S. Pearson, Queens On A Chessboard

Wikipedia, Eight Queens Puzzle

FORMULA

Lim_{n->infinity} Sum_{k=1..n} T(n,k)*x^k = A065188(x).

EXAMPLE

Triangle begins:

n\k  [1]  [2]  [3]  [4]  [5]  [6]  [7]  [8]  [9]  [10] [11] [12]

[1]  1;

[2]  0,   0;

[3]  0,   0,   0;

[4]  2,   4,   1,   3;

[5]  1,   3,   5,   2,   4;

[6]  2,   4,   6,   1,   3,   5;

[7]  1,   3,   5,   7,   2,   4,   6;

[8]  1,   5,   8,   6,   3,   7,   2,   4;

[9]  1,   3,   6,   8,   2,   4,   9,   7,   5;

[10] 1,   3,   6,   8,   10,  5,   9,   2,   4,   7;

[11] 1,   3,   5,   7,   9,   11,  2,   4,   6,   8,   10;

[12] 1,   3,   5,   8,   10,  12,  6,   11,  2,   7,   9,   4;

[13] ...

For n=8 the lexicographically smallest solution for the 8-queens problem is 1,5,8,6,3,7,2,4.

CROSSREFS

Cf. A140450, A000170.

Sequence in context: A254076 A257164 A190555 * A130266 A261595 A211197

Adjacent sequences:  A141840 A141841 A141842 * A141844 A141845 A141846

KEYWORD

nonn,tabl,changed

AUTHOR

Colin S. Pearson

EXTENSIONS

We extended this sequence by adding new terms 1036 to 1128 relating to two further puzzle solutions; for board size 46 X 46 (a new solution) and for board size 47 X 47. Given that the k-th queen is placed in square (k, a(n, k)), we have added the terms (1, a(46, 1)) to (47, a(47, 47)).

Comments rewritten by Matthias Engelhardt, Jan 28 2018

STATUS

approved

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Last modified April 26 03:48 EDT 2018. Contains 303098 sequences. (Running on oeis4.)