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

1,1

Comments

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

References

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

Links

Table of n, a(n) for n=1..105.

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.

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.

Example

2.309138593330494731098720305017212531911814472581628401694402900284456440748...

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]

Crossrefs

Cf. A006958.

Sequence in context: A120473 A019911 A173344 * A020823 A021437 A074760

Adjacent sequences: A276991 A276992 A276993 * A276995 A276996 A276997

Keyword

nonn,cons

Author

Vaclav Kotesovec, Sep 27 2016

Status

approved