This is the decimal expansion of t = 1/rho(7) = 2 + rho(7) - rho(7)^2 with rho(7) = 2*cos(2*Pi/7) the length ratio of the smaller diagonal and the side of a regular heptagon. See A160389 for the decimal expansion of rho(7).

t satisfies the cubic equation t^3 - 2*t^2 - t + 1 = 0.

t = 1/rho(7) is the slope tan(alpha) appearing in Archimedes's neusis construction of the regular heptagon. The corresponding angle alpha is approximately 29,028 degrees. See the link, Figure 1, also for references.