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, 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.

Links

Table of n, a(n) for n=0..40.

R. P. Stanley, Extremal [Chess] Problems

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-12,0,9,0,-2).

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

Adjacent sequences: A078990 A078991 A078992 * A078994 A078995 A078996

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