A242071 Decimal expansion of 'beta', a constant appearing in the random links Traveling Salesman Problem.

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

Offset: 1

Author

Jean-François Alcover, Aug 14 2014

Keywords

• nonn
• cons

Examples

			2.041548186412132418045490158381455866340250252564691915512131281...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.5 Traveling Salesman constants, p. 499.

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 60.

Crossrefs

Cf. A073008, A091505, A240717.

Programs

Mathematica

y[x_] := -2 - ProductLog[-1, E^(-2-x)*(2 - 2*E^x + x)]; beta = (1/2)*NIntegrate[y[x], {x, 0, Infinity}, WorkingPrecision -> 102]; beta // RealDigits // First

Formula

beta = integral_{x>0} y(x) dx, where y(x) = -2 - W_(-1) (e^(-2-x) *(2-2*e^x+x)), W_k(z) being the k-th order Lambert W function (also known as ProductLog). y(x) is implicitly defined by the equation (1+x/2)*exp(-x)+(1+y(x)/2)*exp(-y(x)) = 1.