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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

%I

%S 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,

%T 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,

%U 3,5,8,10,12,6,11,2,7,9,4,1,3,5,2,9,12,10

%N 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)).

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

%C 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.

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

%C 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.

%C 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.

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

%H Matthias Engelhardt, <a href="/A141843/b141843.txt">Table of n, a(n) for n = 1..1892</a>, (previous version of Colin Pearson enlarged)

%H Matthias R. Engelhardt, <a href="http://nqueens.de/sub/SearchAlgorithm.en.html">The old nQueens problem</a>

%H Matteo Fischetti, Domenico Salvagnin, <a href="https://www.researchgate.net/profile/Matteo_Fischetti/publication/322508723_Chasing_First_Queens_by_Integer_Programming/links/5a5cf3620f7e9b4f78396d0b/Chasing-First-Queens-by-Integer-Programming.pdf">Chasing First Queens by Integer Programming</a>, 2018.

%H Colin S. Pearson, <a href="http://queens.cspea.co.uk/">CSP Queens - Counting Queen-placements</a>

%H Martin S. Pearson, <a href="http://queens.lyndenlea.info/">Queens On A Chessboard</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Eight_queens_puzzle">Eight Queens Puzzle</a>

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

%e Triangle begins:

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

%e [1] 1;

%e [2] 0, 0;

%e [3] 0, 0, 0;

%e [4] 2, 4, 1, 3;

%e [5] 1, 3, 5, 2, 4;

%e [6] 2, 4, 6, 1, 3, 5;

%e [7] 1, 3, 5, 7, 2, 4, 6;

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

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

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

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

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

%e [13] ...

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

%Y Cf. A140450, A000170.

%K nonn,tabl

%O 1,7

%A _Colin S. Pearson_

%E 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)).

%E Comments rewritten by _Matthias Engelhardt_, Jan 28 2018

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Last modified June 23 14:02 EDT 2018. Contains 305709 sequences. (Running on oeis4.)