A276994 Decimal expansion of the Klarner-Rivest polyomino constant.

2, 3, 0, 9, 1, 3, 8, 5, 9, 3, 3, 3, 0, 4, 9, 4, 7, 3, 1, 0, 9, 8, 7, 2, 0, 3, 0, 5, 0, 1, 7, 2, 1, 2, 5, 3, 1, 9, 1, 1, 8, 1, 4, 4, 7, 2, 5, 8, 1, 6, 2, 8, 4, 0, 1, 6, 9, 4, 4, 0, 2, 9, 0, 0, 2, 8, 4, 4, 5, 6, 4, 4, 0, 7, 4, 8, 3, 1, 6, 8, 4, 2, 7, 1, 7, 2, 8, 1, 6, 1, 5, 7, 7, 4, 4, 1, 2, 1, 7, 4, 3, 7, 4, 6, 1

Offset: 1

Author

Vaclav Kotesovec, Sep 27 2016

Keywords

• nonn
• cons

Comments

Analytic Combinatorics (Flajolet and Sedgewick, 2009, p. 662) has a wrong value of this constant (2.309138593331230...).

Examples

			2.309138593330494731098720305017212531911814472581628401694402900284456440748...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.19 (Klarner's polyomino constant), p. 380.

Links

• E. A. Bender, Convex n-ominoes, Discrete Math., 8 (1974), 219-226.
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, p. 662.
• D. A. Klarner and R. L. Rivest, Asymptotic bounds for the number of convex n-ominoes, Discrete Math., 8 (1974), 31-40.

Crossrefs

Cf. A006958.

Programs

Mathematica

1/z/.FindRoot[Sum[(-1)^n * z^(n*(n+1)/2) / QPochhammer[z, z, n]^2, {n, 0, 1000}], {z, 2/5}, WorkingPrecision -> 120]

Formula

Equals lim n -> infinity A006958(n)^(1/n).

1/A276994 = 0.4330619231293906645846169654189837... is the smallest positive root of the equation Sum_{n>=0} ((-1)^n * z^(n*(n+1)/2) / (Product_{k=1..n} 1-z^k)^2) = 0.