The Sum of ten consecutive Triangular Numbers:

Starting with T₀ = 0, in base 10,

The sum of the first 10 triangular numbers from T₀ to T₉ = 165
The sum of the next 10 triangular numbers from T₁₀ to T₁₉ = 1165
The sum of the next 10 triangular numbers from T₂₀ to T₂₉ = 3165
The sum of the next 10 triangular numbers from T₃₀ to T₃₉ = 6165
The sum of the next 10 triangular numbers from T₄₀ to T₄₉ = 10165
The sum of the next 10 triangular numbers from T₅₀ to T₅₉ = 15165

and so on.

The same pattern is evident in other bases [when summing T₀ to T_base-1 and so on].

• As submitted by Julian Beauchamp, 9v19, to Shyam Sunder Gupta’s “Fascinating Triangular Numbers”.