## (2*k + 1)^2 * T(n) + T(k) is a triangular number

if   $T_{n}$   is the n-th triangular number then

$(2 \, k + 1)^2 \times T_{n} \; + \; T_{k}$   is a triangular number

$(2 \, k + 1)^2 \; \times \; n \,(n + 1)/2 \; + \; k \,(k + 1)/2$

$= \; 1/2 \; (4 \, k^2 \, n^2 + 4 \, k^2 \, n + k^2 + 4 \, k \, n^2 + 4 \, k \, n + k + n^2 + n)$

$= \; 1/2 \; (2 \, k \, n + k + n) \, (2 \, k \, n + k + n + 1)$

Hence,

$(2 \, k + 1)^2 \times T_{n} \; + \; T_{k} \; = \; T_{ \,2 \, k \, n + k + n \,}$