A246952 Decimal expansion of sigma, a constant appearing in the asymptotic expression of the number a(n) of Carlitz compositions of size n.

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

Offset

0,1

Links

Table of n, a(n) for n=0..102.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 38.

A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Formula

Sigma is the unique solution of the equation F(x)=1, 0 <= x <= 1, where F(x) = sum_{j>=1} (-1)^(j - 1)*(x^j/(1 - x^j)).

a(n) ~ 1/(sigma*F'(sigma))*(1/sigma)^n = c*r^n, where c = 0.456387... and r = A241902 = 1.750243...

Example

0.571349793158087643112217904891974600336176224937534145...

Mathematica

digits = 103; F[x_?NumericQ] := NSum[(-1)^(j - 1)*(x^j/(1 - x^j)), {j, 1, Infinity}, WorkingPrecision -> digits+5]; sigma = x /. FindRoot[F[x] == 1, {x, 2/5, 1/2}, WorkingPrecision -> digits+5]; RealDigits[sigma, 10, digits] // First

Crossrefs

Cf. A003242, A241902.

Sequence in context: A195300 A019697 A217173 * A323386 A021179 A153613

Adjacent sequences: A246949 A246950 A246951 * A246953 A246954 A246955

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 08 2014

Status

approved