Strong empirical evidence supports conjectures that certain number patterns always hold. These patterns concern the function cr, defined by the equation cr(n) = n - m2, m2 being the nearest square to n, on the domain of the triangular numbers. Triangular squares or triangular numbers of the form m2+m are also mentioned in most of the conjectures. One of the conjectures, for example, is that the sum of cr over the triangular numbers up to a triangular square is 0. Some of these patterns can be described by strings of symbols, such as "S" and "L," formed by first writing one symbol and then simultaneously substituting "SLSLS" for "S" and "SLSLSLS" for "L," and continuing to make that substitution indefinitely.
Some Conjectures Concerning Triangular Numbers.
Journal of the Minnesota Academy of Science, Vol. 58 No.2, 21-25.
Retrieved from https://digitalcommons.morris.umn.edu/jmas/vol58/iss2/4