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A085801
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Maximum number of nonattacking queens on an n X n toroidal board.
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5
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1, 1, 1, 2, 5, 4, 7, 6, 7, 9, 11, 10, 13, 13, 13, 14, 17, 16, 19, 18, 19, 21, 23, 22, 25, 25, 25, 26, 29, 28, 31, 30, 31, 33, 35, 34, 37, 37, 37, 38, 41, 40, 43, 42, 43, 45, 47, 46, 49, 49, 49, 50, 53, 52, 55, 54, 55, 57, 59, 58
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Polya showed that a(n)=n if n is not divisible by 2 and not divisible by 3. Schlude and Specker showed the following: If n is even but not divisible by 3 and not divisible by 4, then a(n)=n-1. If n is divisible by 3, then a(n)<n-1. If n is divisible by 4 but not by 8, then a(n)<n-1. There are open questions as well: Does it hold that a(n)>n-3 for every n? Does it hold that a(n)<n-1 if n is divisible by 8?
Comment from Franklin T. Adams-Watters, Feb 06 2006: According to the Cairns reference, it appears that the questions above were solved by Monsky.
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REFERENCES
| Eldar Fischer, Tomer Kotek, and Johann A. Makowsky, Application of Logic to Combinatorial Sequences and Their Recurrence Relations, http://www.cs.technion.ac.il/~tkotek/pubfiles/12-FKM-final.pdf
G. Polya: Ueber die 'Doppelt-Periodischen' Loesungen des n-Damen-Problems, in: W. Ahrens: Mathematische Unterhaltungen und Spiele, Teubner, Leipzig, 1918, 364-374. Reprinted in: G. Polya: Collected Works, Vol. V, 237-247.
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LINKS
| Grant Cairns, Queens on Non-square Tori, El. J. Combinatorics, N6, 2001
P. Monsky, Problem E3162, Amer. Math. Monthly 96 (1989), 258-259.
Konrad Schlude and Ernst Specker, Zum Problem der Damen auf dem Torus, Technical Report 412, Computer Science Department ETH Zurich, 2003.
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FORMULA
| a(n) = n, if n is not divisible by 2 and not by 3. a(n) = n-1, if n is divisible by 2, but not by 3 and not by 4 a(n) < n-1, if n is divisible by 3
(Based on the first 1000 terms:) G.f. (2*x^12 - x^11 + 2*x^10 + 2*x^9 + x^8 - x^7 + 3*x^6 - x^5 + 3*x^4 + x^3 + 1)/(x^13 - x^12 - x + 1) =
(2*x^12 - x^11 + 2*x^10 + 2*x^9 + x^8 - x^7 + 3*x^6 - x^5 + 3*x^4 + x^3 + 1)/((x - 1)^2*(x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^2 + x + 1)*(x^4 - x^2 + 1)) . Joerg Arndt, Dec 13 2010
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EXAMPLE
| a(5)=5 because 5 is not divisible by 2 and not divisible by 3
a(10)=10-1=9 because 10 is divisible by 2, but not by 3 and not by 4
a(12)=12-2=10 because 12 is divisible by 3
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MATHEMATICA
| (* Explicit formula, based on a article by Monsky, Vaclav Kotesovec, Dec 13 2010 *)
Table[n-1/6*(2*Cos[Pi*n/2]-3*Cos[Pi*n/3]+5*Cos[2*Pi*n/3]-Cos[Pi*n/6]-Cos[5*Pi*n/6]+3*Cos[Pi*n]+7), {n, 1, 100}]
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PROG
| (PARI) a(n)=n-1/6*(2*cos(Pi*n/2)-3*cos(Pi*n/3)+5*cos(2*Pi*n/3)-cos(Pi*n/6)-cos(5*Pi*n/6)+3*cos(Pi*n)+7);
vector(60, n, round(a(n))) \\ Joerg Arndt, Dec 13 2010
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CROSSREFS
| Cf. A051906, A007705.
Sequence in context: A102513 A100116 A107921 * A023843 A153990 A154811
Adjacent sequences: A085798 A085799 A085800 * A085802 A085803 A085804
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KEYWORD
| easy,nonn
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AUTHOR
| Konrad Schlude (schlude(AT)inf.ethz.ch), Jul 24 2003
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