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

Author

Jean-François Alcover, Sep 08 2014

Keywords

• nonn
• cons

Examples

			0.571349793158087643112217904891974600336176224937534145...
		

Links

• 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.

Crossrefs

Cf. A003242, A241902.

Programs

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

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...