Integer as Sum of Polygonal Numbers/Lemma 1

Theorem
Let $n, m \in \R_{>0}$ such that $\dfrac n m > 1$.

Let $n < 116 m$.

Then $n$ can be expressed as a sum of at most $m + 2$ polygonal numbers of order $m + 2$.

Proof
From Closed Form for Polygonal Numbers:


 * $\map P {m + 2, k} = \dfrac m 2 \paren {k^2 - k} + k = m T_{k - 1} + k$

Where $T_{k - 1}$ are triangular numbers.

The first few $\paren {m + 2}$-gonal numbers less than $116 m$ are:


 * $0, 1, m + 2, 3 m + 3, 6 m + 4, 10 m + 5, 15 m + 6, 21 m + 7, 28 m + 8, 36 m + 9, 45 m + 10, 55 m + 11, 66 m + 12, 78 m + 13, 91 m + 14, 105 m + 15$

We show the expression of the first few numbers explicitly.

and so on.