cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A173775 Number of ways to place 5 nonattacking queens on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 10, 0, 882, 13312, 85536, 561440, 2276736, 9471744, 27991470, 85725696, 209107890, 525062144, 1116665944, 2437807104, 4691672964, 9234168960, 16462896030, 29919532544, 50215537658, 85687824384, 136944081500
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 24 2010

Keywords

Links

Crossrefs

Cf. A172517, A172518, A172519, A007705, A108792, A061991.

Formula

a(n) = (1/120)*n^10 - (1/3)*n^9 + (143/24)*n^8 - (373/6*n^7) + (99377/240)*n^6 - (3603/2)*n^5 + (119627/24)*n^4 - (23833/3)*n^3 + (16342/3)*n^2 + ((1/24)*n^8 - (3/2)*n^7 + (1111/48)*n^6 - (391/2)*n^5 + (7595/8)*n^4 - 2487*n^3 + (8032/3)*n^2)*(-1)^n + ((9/2)*n^4 - 78*n^3 + 374*n^2)*cos(Pi*n/2) + ((8/3)*n^4 - (128/3)*n^3 + (656/3)*n^2)*cos(2*Pi*n/3) + (80/3)*n^2*cos(Pi*n/3) + (16/5)*n^2*cos(2*Pi*n/5) + (16/5)*n^2*cos(Pi*n/5)*(-1)^n.
Recurrence: a(n) = -3a(n-1) - 5a(n-2) - 5a(n-3) + 2a(n-4) + 17a(n-5) + 37a(n-6) + 49a(n-7) + 35a(n-8) - 16a(n-9) - 101a(n-10) - 185a(n-11) - 215a(n-12) - 139a(n-13) + 56a(n-14) + 321a(n-15) + 544a(n-16) + 588a(n-17) + 368a(n-18) - 99a(n-19) - 656a(n-20) - 1069a(n-21) - 1111a(n-22) - 689a(n-23) + 84a(n-24) + 929a(n-25) + 1488a(n-26) + 1506a(n-27) + 939a(n-28) - 939a(n-30) - 1506a(n-31) - 1488a(n-32) - 929a(n-33) - 84a(n-34) + 689a(n-35) + 1111a(n-36) + 1069a(n-37) + 656a(n-38) + 99a(n-39)-368a(n-40) - 588a(n-41) - 544a(n-42) - 321a(n-43) - 56a(n-44) + 139a(n-45) + 215a(n-46) + 185a(n-47) + 101a(n-48) + 16a(n-49) - 35a(n-50) - 49a(n-51) - 37a(n-52) - 17a(n-53) - 2a(n-54) + 5a(n-55) + 5a(n-56) + 3a(n-57) + a(n-58).

A054500 Indicator sequence for classification of nonattacking queens on n X n toroidal board.

Original entry on oeis.org

1, 5, 7, 11, 13, 13, 13, 13, 17, 17, 17, 17, 17, 19, 19, 19, 23, 23, 23, 25, 25, 25, 25, 25, 25, 25, 25, 29, 29, 29, 29, 29
Offset: 1

Views

Author

Matthias Engelhardt

Keywords

Comments

The three sequences A054500/A054501/A054502 are used to classify solutions to the problem of "Nonattacking queens on a 2n+1 X 2n+1 toroidal board" by their symmetry; solutions are considered equivalent iff they differ only by rotation, reflection or torus shift.
For brevity, let i(n) = A054500(n) (indicator sequence), m(n) = A054501(n) (multiplicity) and c(n) = A054502(n) (count).
i(n) = k means that there are solutions for the k X k board and that m(n) and c(n) refer to it. There are c(n) inequivalent solutions which may be extended to m(n) different representations each (i.e., m(n) permutations).
This gives two formulas: A007705(n) = Sum (c(k) * m(k)), A053994(n) = Sum (c(k)), where the sum is taken over all k for which i(k) = 2n+1, for both formulas. Note that m(n) is always a divisor of 8 * i(n)^2.

Examples

			For a 19 X 19 toroidal board, you have three entries in the indicator sequence A054500; their count terms (A054502) give 354 = 4 + 132 + 218 inequivalent solutions; together with their multiplicity (A054501) they add up to 4*76 + 132*1444 + 218*2888 = 820496 solutions at all.

References

  • A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982 (for getting equivalence classes).

Links

  • Manuel Kauers and Christoph Koutschan, Guessing with Little Data, arXiv:2202.07966 [cs.SC], 2022.
  • I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math.Monthly, 101 (1994), 629-639 (for finding the solutions).

Crossrefs

Cf. A054501, A054502, A053994, A007705, A006841.

Extensions

More terms from Matthias Engelhardt, Jan 11 2001

A181499 Triangle read by rows: number of solutions of n queens problem for given n and given number of conflicts.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 0, 0, 8, 4, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 0, 96, 24, 0, 0, 0, 0, 0, 0, 240, 0, 372, 112, 0, 0, 0, 0, 0, 88, 0, 0, 328, 1252, 872, 140, 0, 0, 0, 0, 0, 0, 0, 0, 3016, 5140, 4696, 1316, 32, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Matthias Engelhardt, Oct 25 2010

Keywords

Examples

			For n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two conflicts So the terms for n=4 are 0 (0 solutions for n=4 having 0 conflicts), 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

Links

Crossrefs

Cf. A000170, A007705, A181500, A181501, A181502.

Formula

Row sum = A000170 (number of n queens placements)
Column 0 has same values as A007705 (torus n queens solutions)
Column 1 is always zero.

A181500 Triangle read by rows: number of solutions of n queens problem for given n and given number of queens engaged in conflicts.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 28, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 64, 0, 28, 0, 0, 0, 0, 0, 0, 232, 8, 32, 48, 32
Offset: 0

Views

Author

Matthias Engelhardt, Oct 30 2010

Keywords

Comments

Schlude and Specker investigate if it is possible to set n-1 non-attacking queens on an n X n toroidal chessboard. That is equivalent to searching for normal (i.e., non-toroidal) solutions of 3 engaged queens. In this case, one of the three queens has conflicts with both other queens. If you remove this queen, you get a setting of n-1 queens without conflicts, i.e., a toroidal solution.

Examples

			Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 0, 0, 2;
  10, 0, 0, 0, 0, 0;
   0, 0, 0, 0, 4, 0,  0;
  28, 0, 0, 0, 0, 0, 12, 0;
... - _Andrew Howroyd_, Dec 31 2017
For n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. For both solutions, all 4 queens are engaged in conflicts. So the terms for n=4 are 0 (0 solutions for n=4 having 0 engaged queens), 0, 0, 0 and 2 (the two cited above). These are members 11 to 15 of the sequence.

Links

Crossrefs

Cf. A181499, A181501, A181502.

Formula

Row sum = A000170 (number of n-queen placements).
Column 0 has same values as A007705 (torus n-queen solutions).
Columns 1 and 2 are always zero.
Column 3 counts solutions of the special "Schlude-Specker" situation.

Extensions

Offset corrected by Andrew Howroyd, Dec 31 2017

A181501 Triangle read by rows: number of solutions of n queens problem for given n and given number of connection components of conflict constellation.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 10, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 28, 0, 4, 8, 0, 0, 0, 0, 0, 0, 92, 0, 0, 0, 0, 0, 0, 0, 8, 272, 56, 16, 0, 0, 0, 0, 0, 0, 96, 344, 240, 44, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Matthias Engelhardt, Oct 30 2010

Keywords

Comments

The rightmost part of the triangle contains only zeros. As any connection component needs at least two queens, the number of connection components of a solution is always less than or equal to n.

Examples

			Triangle begins:
   0;
   1, 0;
   0, 0, 0;
   0, 0, 0, 0;
   0, 0, 2, 0, 0;
  10, 0, 0, 0, 0, 0;
   0, 4, 0, 0, 0, 0, 0;
  28, 0, 4, 8, 0, 0, 0, 0;
  ... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two connection components in the conflicts graph. So, the terms for n=4 are 0, 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

Links

Crossrefs

Cf. A000170, A007705, A181499, A181500, A181502.

Formula

Row sum =A000170 (number of n queens placements)
Column 0 has same values as A007705 (torus n queens solutions)

Extensions

Offset corrected by Andrew Howroyd, Dec 31 2017

A181502 Triangle read by rows: number of solutions of n queens problem for given n and given maximal size of a connection component in the conflict constellation.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 28, 8, 4, 0, 0, 0, 0, 0, 0, 64, 24, 4, 0, 0, 0, 0, 0, 0, 248, 80, 16, 8, 0, 0, 0, 0, 0, 0, 172, 484, 36, 32, 0, 0, 0
Offset: 0

Views

Author

Matthias Engelhardt, Oct 30 2010

Keywords

Comments

Torus solutions, i.e. solutions having an empty conflict constellation, are counted in column 1; this is caused by an interpretation of a queen not engaged in any conflict as an island in the conflict graph. Using the definition strictly, these queens should be removed from the graph and the numbers should appear in column 0, not column 1.

Examples

			Triangle begins:
  0;
  0,  1;
  0,  0, 0;
  0,  0, 0, 0;
  0,  0, 2, 0, 0;
  0, 10, 0, 0, 0, 0;
  0,  0, 0, 0, 4, 0, 0;
  0, 28, 8, 4, 0, 0, 0, 0;
... - _Andrew Howroyd_, Dec 31 2017
for n=4, there are only the two solutions 2-4-1-3 and 3-1-4-2. Both have two conflicts So the terms for n=4 are 0 (0 solutions for n=4 having 0 conflicts), 0, 2 (the two cited above), 0 and 0. These are members 10 to 15 of the sequence.

Links

Crossrefs

Cf. A000170, A007705, A181499, A181500, A181501.

Formula

Row sum =A000170 (number of n queens placements)
Column 1 has same values as A007705 (torus n queens solutions)
Column 0 is always zero.

Extensions

Offset corrected by Andrew Howroyd, Dec 31 2017

A054501 Multiplicity sequence for classification of nonattacking queens on n X n toroidal board.

Original entry on oeis.org

1, 10, 28, 44, 26, 52, 338, 676, 34, 68, 578, 1156, 2312, 76, 1444, 2888, 92, 2116, 4232, 50, 100, 250, 500, 1000, 1250, 2500, 5000, 58, 116, 1682, 3364, 6728
Offset: 1

Views

Author

Matthias Engelhardt

Keywords

Crossrefs

See comments and references for A054500.
Cf. A054500, A054502, A053994, A007705, A006841.

Extensions

More terms from Matthias Engelhardt, Jan 11 2001

A054502 Counting sequence for classification of nonattacking queens on n X n toroidal board.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 5, 1, 3, 23, 30, 40, 4, 132, 218, 5, 1859, 29517, 1, 2, 9, 18, 51, 470, 7170, 387830, 1, 6, 1215, 121487, 89997968
Offset: 1

Views

Author

Matthias Engelhardt

Keywords

Crossrefs

See comments and references for A054500.
Cf. A054500, A054501, A053994, A007705, A006841.

Extensions

More terms from Matthias Engelhardt, Jan 11 2001

A062164 Number of ways of placing n nonattacking (normal) queens on n X n board; solutions congruent on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 6, 20, 40, 191, 953, 4604, 24660, 158466, 1009395
Offset: 1

Views

Author

Matthias Engelhardt

Keywords

Comments

In this sequence two n-queens solutions p and q are considered equivalent iff there are natural numbers x and y such that, for all k from {0, ..., n-1}, q (k + x mod n) = p (k) + y mod n, or q is a rotation or a reflection of such a q.
In other words, besides rotations and reflections, also torus shifts are allowed. The sequence reduces the objects of A002562 and via that of A000170. The reduction of A000170 to this sequence is exactly the same as from A007705 to A053994 for torus queens; however, a solution for torus queens remains always a solution after a shift while a normal queens solutions does so only sometimes.
Note that the equivalence classes of this sequence are a subset of A006841. Moreover they are a subset of A062167.

Links

Extensions

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010

A062167 Number of permutations with at most 2 queens on any torus diagonal, solutions congruent on the torus count only once.

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 5, 29, 93, 569, 3226, 28630, 221250, 2314650
Offset: 1

Views

Author

Matthias Engelhardt

Keywords

Comments

This sequence counts classes of "near n-queens solutions". Permutations with at most 1 queen on any torus diagonal are exactly the torus n queen solutions (A007705), those with at most 2 contain the normal n queen solutions (A000170).
Therefore they may be called "near n-queens solutions". In this sequence, permutations p and q are considered equivalent iff there are natural x and y, such that, for all k from {0, ..., n-1}, q (k + x mod n) = p (k) + y mod n, or q is a rotation or a reflection of such a q. In other words, rotations, reflections and torus shifts are allowed. The sequence contains the objects of A062164.

Links

Extensions

Updated link that is transferred from people.freenet.de/nQueens to www.nqueens.de Matthias Engelhardt, Apr 21 2010
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