By definition, triangular number T_n is the sum

T_n=1+2+3+\ldots+n.

1. T_n+(n+1)=T_{n+1}

By definition.

2. T_n = \frac{n(n+1)}{2} = n\choose{2}

By induction.

3. T_n + T_{n-1} = (T_n - T_{n-1})^2

\frac{n(n+1)}{2}+\frac{(n-1)n}{2}=n^2=\Big(\frac{(n+1)n}{2}-\frac{(n-1)n}{2}\Big)^2.

4. T_{n}^{2}+T_{n-1}^{2}=T_{n^{2}}

\frac{n^{2}(n+1)^{2}}{2}+\frac{n^{2}(n-1)^{2}}{2}=\frac{n^{2}(2n^{2}+2)}{4}=\frac{n^{2}(n^{2}+1)}{2}.

5. T_{n+1}-T_{n-1}=T_{2n+1}-T_{2n}

6. 3T_n+T_{n-1}=T_{2n}

7. 3T_n+T_{n+1}=T_{2n+1}

8. T_{n+1}^{2}-T_{n}^{2}=(n+1)^{3}

9. \sum^{2n-1}_{k=1}(-1)^{k+1}T_{k}=n^2

10. n\cdot T_{n+1}=(n+2)\cdot T_{n}

11. T_{n}T_{k}+T_{n-1}T_{k-1}=T_{nk}

12. T_{n}T_{k-1}+T_{n-1}T_{k}=T_{nk-1}

13. n^{2}T_{k-1}+kT_{n}=T_{nk}

14. n^{2}T_{k-1}+kT_{n-1}=T_{nk-1}

15. T_{n-1}+6T_{n}+T_{n+1}=(2n+1)^2

16. (2k+1)^{2}T_{n}+T_{k}=T_{(2k+1)n+k}