Smallest Number Expressible as Sum of at most Three Triangular Numbers in 4 ways

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Theorem

$21$ is the smallest number which can be expressed as the sum of at most $3$ triangular numbers in $4$ ways.


Proof

By inspection:

\(\ds 1\) \(=\) \(\ds T_1\) $1$ way
\(\ds 2\) \(=\) \(\ds T_1 + T_1\) $1$ way
\(\ds 3\) \(=\) \(\ds T_1 + T_1 + T_1\)
\(\ds \) \(=\) \(\ds T_2\) $2$ ways
\(\ds 4\) \(=\) \(\ds T_2 + T_1\) $1$ way
\(\ds 5\) \(=\) \(\ds T_2 + T_1 + T_1\) $1$ way
\(\ds 6\) \(=\) \(\ds T_2 + T_2\)
\(\ds \) \(=\) \(\ds T_3\) $2$ ways
\(\ds 7\) \(=\) \(\ds T_2 + T_2 + T_1\)
\(\ds \) \(=\) \(\ds T_3 + T_1\) $2$ ways
\(\ds 8\) \(=\) \(\ds T_3 + T_1 + T_1\) $1$ way
\(\ds 9\) \(=\) \(\ds T_3 + T_2\)
\(\ds \) \(=\) \(\ds T_2 + T_2 + T_2\) $2$ ways
\(\ds 10\) \(=\) \(\ds T_4\)
\(\ds \) \(=\) \(\ds T_3 + T_2 + T_1\) $2$ ways
\(\ds 11\) \(=\) \(\ds T_4 + T_1\) $1$ way
\(\ds 12\) \(=\) \(\ds T_4 + T_1 + T_1\)
\(\ds \) \(=\) \(\ds T_3 + T_3\)
\(\ds \) \(=\) \(\ds T_3 + T_2 + T_2\) $3$ ways
\(\ds 13\) \(=\) \(\ds T_4 + T_2\)
\(\ds \) \(=\) \(\ds T_3 + T_3 + T_1\) $2$ ways
\(\ds 14\) \(=\) \(\ds T_4 + T_2 + T_1\) $1$ way
\(\ds 15\) \(=\) \(\ds T_5\)
\(\ds \) \(=\) \(\ds T_3 + T_3 + T_2\) $2$ ways
\(\ds 16\) \(=\) \(\ds T_5 + T_1\)
\(\ds \) \(=\) \(\ds T_4 + T_3\)
\(\ds \) \(=\) \(\ds T_4 + T_2 + T_2\) $3$ ways
\(\ds 17\) \(=\) \(\ds T_5 + T_1 + T_1\)
\(\ds \) \(=\) \(\ds T_4 + T_3 + T_1\) $2$ ways
\(\ds 18\) \(=\) \(\ds T_5 + T_2\)
\(\ds \) \(=\) \(\ds T_3 + T_3 + T_3\) $2$ ways
\(\ds 19\) \(=\) \(\ds T_5 + T_2 + T_1\)
\(\ds \) \(=\) \(\ds T_4 + T_3 + T_2\) $2$ ways
\(\ds 20\) \(=\) \(\ds T_4 + T_4\) $1$ way
\(\ds 21\) \(=\) \(\ds T_6\)
\(\ds \) \(=\) \(\ds T_5 + T_3\)
\(\ds \) \(=\) \(\ds T_5 + T_2 + T_2\)
\(\ds \) \(=\) \(\ds T_4 + T_4 + T_1\) $4$ ways

This sequence is A002636 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$


Sources

in which there are many errata
in which the errata in the above are resolved