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A078993
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Starting at the chess position shown, a(n) is the number of ways Black can make n consecutive moves, followed by a checkmate in one move by White.
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1
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0, 0, 0, 0, 0, 2, 5, 8, 28, 24, 108, 66, 357, 176, 1088, 464, 3160, 1218, 8901, 3192, 24564, 8360, 66836, 21890, 180037, 57312, 481464, 150048, 1280736, 392834, 3393509, 1028456, 8965324, 2692536, 23633532, 7049154, 62197413, 18454928, 163482992, 48315632, 429300136
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OFFSET
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0,6
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COMMENTS
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Starting position: White queen at g8, king at h1; Black pawn at h7, king at h6. Black may not move into check.
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REFERENCES
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Problem composed by N. D. Elkies.
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LINKS
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Table of n, a(n) for n=0..40.
R. P. Stanley, Extremal [Chess] Problems
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FORMULA
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G.f.: sum(a(n)*x^n, n=0..infinity) = x^5*(2+5*x-4*x^2-2*x^3)/((1-x^2)*(1-2*x^2)*(1-3*x^2+x^4)).
a(2*n) = 3 - 2^(n+2) + F(2*n+3) for n>0 and a(2*n+1) = 2*(F(2*n-1)-1) with F(n) the Fibonacci numbers.
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EXAMPLE
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For n = 5 we have the move orders: (1): 1.Kh5 2.Kh4 3.Kh3 4.h5 5.h4; (2): 1.Kh5 2.Kh4 3.h5 4.Kh3 5.h4; both followed by Qg2# and a(5) = 2.
For n = 6 we have the move orders: (1): 1.Kh5 2.Kh4 3.Kh3 4.h6 5.h5 6.h4; (2): 1.Kh5 2.Kh4 3.h6 4.h5 5.Kh3 6.h4; (3): 1.Kh5 2.Kh4 3.h6 4:Kh3 5.h5 6.h4; (4): 1.Kh5 2.h6 3.Kh4 4.Kh3 5.h5 6.h4; (5): 1.Kh5 2.h6 3.Kh4 4.h5 5.Kh3 6.h4; all followed by Qg2# and a(6) = 5.
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CROSSREFS
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Cf. A000045 (Fibonacci), A027941 (Fibonacci(2*n+1)-1).
Sequence in context: A093365 A209865 A128600 * A066846 A140275 A025533
Adjacent sequences: A078990 A078991 A078992 * A078994 A078995 A078996
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Jan 18 2003
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EXTENSIONS
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Formula corrected, examples, formulae and crossrefs added and edited by Johannes W. Meijer, Feb 06 2010 and Feb 8 2010.
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STATUS
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approved
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