The Thue number is not clearly related to your number. The decimals with 1s in positions that are n(n+1)/2 or n^2, etc., are related to the Jacobi identity and, in particular,
Product [0 to infinity] {(1+x^n)(1-x^(2n+2))} = 2Sum [x^(n(n+1)/2)]
{I am having a little trouble seeing the...