Representing an Integer as a Sum of at Most $k$ Triangular Numbers What is the smallest $k$ such that every $n \in \mathbb{N}$ can be represented by a sum of exactly $k$ triangular numbers?  For the sake of simplicity, I will assume $0$ is a triangular number.
I've been able to establish that $3 \le k \le 8$.  The lower bound was found through experimentation.  The upper bound is a consequence of Waring's problem, and also the neat property that two consecutive triangular numbers always sum to a square.
Any ideas on how to obtain more reasonable bounds on $k$?  Or perhaps even some obvious method to determine $k$ which I have overlooked?
Related Question(s):


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*Representing a number as a sum of at most $k$ squares.

According to a theorem of Fermat-Legendre-Gauss-Cauchy, any integer can be written as the sum of at most $n$ $n$-gonal numbers. This generalizes Lagrange's theorem on sums of four squares.
(Remark that Fermat's conjecture makes no claim that $n$ is optimal. For instance it is conjectured that every positive integer is the sum of at most 7 octahedral numbers. However, it is easy to see that 3 is optimal for triangular numbers.)
