A002564 -id:A002564 - cp's OEIS frontend

A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

1, 1, 2, 1, 2, 2, 0, 1, -2, 2, 0, 2, 2, 0, 4, 1, 2, -2, 0, 2, 0, 0, 0, 2, 3, 2, 2, 0, 2, 4, 0, 1, 0, 2, 0, -2, 2, 0, 4, 2, 2, 0, 0, 0, -4, 0, 0, 2, 1, 3, 4, 2, 2, 2, 0, 0, 0, 2, 0, 4, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, -2, 2, 2, 6, 0, 0, 4, 0, 2, -2, 2, 0, 0, 4, 0, 4, 0, 2, -4, 0, 0, 0, 0, 0, 2, 2, 1, 0, 3, 2, 4, 0, 2, 0

Offset: 1

Michael Somos, Sep 15 2006

			q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + q^8 - 2*q^9 + 2*q^10 + 2*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002564, A019670, A035154, A113446, A122856, A122865.

eta[x_] := x^(1/24)*QPochhammer[x]; A122864[n_] := SeriesCoefficient[ (eta[q^3]^2*eta[q^4]*eta[q^6]^2*eta[q^36])/(eta[q]*eta[q^9]*eta[q^12]^2), {q, 0, n}]; Table[A122864[n], {n, 50}] (* G. C. Greubel, Sep 16 2017 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^36 + A) /(eta(x + A) * eta(x^9 + A) * eta(x^12 + A)^2), n))}

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, -2*(-1)^e, if( p%4==1, e+1, 1-e%2))))))}

Euler transform of period 36 sequence [ 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, 1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, 1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2*(-1)^(e+1) if e>0, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
a(3*n) = 2 * A113446(n). a(3*n + 1) = A002564(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Oct 15 2022

A002568 Number of different ways one can attack all squares on an n X n chessboard with the smallest number of non-attacking queens needed.

Original entry on oeis.org

1, 4, 1, 16, 16, 120, 8, 728, 92, 8, 2, 840, 24, 436, 10188, 128, 12, 224, 8424, 312, 72, 192, 8784, 368, 56, 224, 14500, 280, 10880, 240

Offset: 1

N. J. A. Sloane

For same problem, but with queens in general position (without condition "non-attacking"), see A002564. - Vaclav Kotesovec, Sep 07 2012

			a(5) = 16 because it is impossible to attack all squares with 2 queens but with 3 queens you can do it in 16 different ways (with mirroring and rotation).

W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Mia Müßig, Julia code to compute the sequence
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49. [Incomplete annotated scan of title page and pages 18-51]

See A002567 for the number of non-isomorphic solutions.

Cf. A002564, A122749, A103315.

a(9)-a(12) from Johan Särnbratt, Mar 28 2008
Name of the sequence corrected by Vaclav Kotesovec, Sep 07 2012
a(13)-a(15) from Andrew Howroyd, Dec 07 2021
a(16)-a(30) from Mia Muessig, Oct 04 2024

A002563 Number of nonisomorphic solutions to minimal dominating set on queens' graph Q(n).

Original entry on oeis.org

1, 1, 1, 3, 37, 1, 13, 638, 21, 1, 1, 1, 41, 588, 25872, 43, 22, 2

Offset: 1

N. J. A. Sloane

W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
W. W. R. Ball and H. S. M. Coxeter, Math'l Rec. and Essays, 13th Ed. Dover, p. 173.
Teresa W. Haynes, Stephen T. Hedetniemi and Michael A. Henning (eds.), Structures of Domination in Graphs, Springer, 2021. See Table 14 on p. 368.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Australasian Journal of Combinatorics 23 (2001), 253-284.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49. [Incomplete annotated scan of title page and pages 18-51]

See A002564 for number of distinct solutions.

A075458 gives number of queens required.

a(16)-a(18) from "Structures of Domination in Graphs" added by Andrey Zabolotskiy, Sep 02 2021

A286883 Number of minimal dominating sets in the n X n queen graph.

Original entry on oeis.org

1, 4, 21, 194, 2579, 48028, 1023698, 28281838

Offset: 1

Eric W. Weisstein, Aug 02 2017

Eric Weisstein's World of Mathematics, Minimal Dominating Set
Eric Weisstein's World of Mathematics, Queen Graph

Cf. A002564, A002568, A075692, A287070, A290711, A291103.

a(5)-a(8) from Andrew Howroyd, Aug 19 2017

A002566 Number of ways to attack all squares on an n X n chessboard using the smallest possible number of queens with each queen attacking at least one other.

Original entry on oeis.org

0, 6, 20, 12, 70, 960, 22, 352, 10, 216, 4814, 72

Offset: 1

N. J. A. Sloane

Differs from A002564 and A002568 in that each queen is attacking at least one other queen.
M. A. Sainte-Laguë paper has "a(6)=900?".
In other words, the number of minimum total dominating sets in the n X n queen graph. - Eric W. Weisstein, Apr 19 2018

W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49. [Incomplete annotated scan of title page and pages 18-51]
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Queen Graph.

Cf. A002565 (number of ways up to isomorphism).

a(6) corrected and a(9)-a(11) by Sean A. Irvine, Apr 05 2014
Better name from Sean A. Irvine, Apr 05 2014
a(12) from Eric W. Weisstein, Apr 05 2025

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A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

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A002568 Number of different ways one can attack all squares on an n X n chessboard with the smallest number of non-attacking queens needed.

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A002563 Number of nonisomorphic solutions to minimal dominating set on queens' graph Q(n).

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A286883 Number of minimal dominating sets in the n X n queen graph.

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A002566 Number of ways to attack all squares on an n X n chessboard using the smallest possible number of queens with each queen attacking at least one other.

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