A228246 a(1)=1; for n >= 2, a(n) = round(x), where x is the average length (not counting draws) of a first-to-n match between two chess players of equal strength.

1, 2, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 29, 31, 33, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 107, 109, 111, 113, 115, 117, 119

Offset

1,2

Comments

A first-to-n match between two chess players continues until either player reaches n wins. For example, in the 1978 and 1981 matches between A. Karpov and V. Korchnoi, n was 6; in the 1992 match between B. Fischer and B. Spassky, n was 10.

In chess, as in some other sports, e.g., football, a score k:m is called "big" if |k-m| >= 3. It is interesting that, e.g., a(6) = 9 corresponds to a big score, 6:3, and a(10) = 16 corresponds to a big score, 10:6. Thus a moderately big score in such matches is not a sufficient reason to revise a relative relation of strengths of the winner and the loser.

References

V. S. Shevelev, Paradox of a big score, Preprint, VINITI, 1992 (in Russian).

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Formula

a(n) = round(2*n*(1 - 4^(-n)*binomial(2*n,n)));

a(n) = 2*n*(1 - 1/sqrt(Pi*n)) + O(1/sqrt(n)).

Mathematica

(Round[2 # (1- Binomial[2 #, #]/4^#)]&)/@Range[20] (* Peter J. C. Moses, Aug 19 2013 *)

Crossrefs

Cf. A000984.

Sequence in context: A047294 A195173 A215000 * A184587 A345436 A225773

Adjacent sequences: A228243 A228244 A228245 * A228247 A228248 A228249

Keyword

nonn

Author

Vladimir Shevelev, Aug 18 2013

Extensions

More terms from Peter J. C. Moses and Giovanni Resta

Status

approved