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A078993
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.
0
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
OFFSET
0,6
COMMENTS
Starting position: White queen at g8, king at h1; Black pawn at h7, king at h6. Black may not move into check.
REFERENCES
Problem composed by N. D. Elkies.
FORMULA
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.
EXAMPLE
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.
MATHEMATICA
LinearRecurrence[{0, 6, 0, -12, 0, 9, 0, -2}, {0, 0, 0, 0, 0, 2, 5, 8, 28}, 50] (* Paolo Xausa, Apr 22 2024 *)
CROSSREFS
Cf. A000045 (Fibonacci), A027941 (Fibonacci(2*n+1)-1).
Sequence in context: A093365 A209865 A128600 * A066846 A304043 A290594
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 18 2003
EXTENSIONS
Formula corrected, examples, formulas and crossrefs added and edited by Johannes W. Meijer, Feb 06 2010 and Feb 08 2010
STATUS
approved