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
- 1946: Gino Loria: Sulla scomposizione di un intero nella somma di numeri poligonali (Atti Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Nat. Rendiconti Ser. 8 Vol. 1: pp. 7 – 15)
- in which there are many errata
- Jul. 1947: D.H. Lehmer: Recent Mathematical Tables (Mathematical Tables and Other Aids to Computation Vol. 2, no. 19: pp. 297 – 311) www.jstor.org/stable/2002589
- in which the errata in the above are resolved
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $21$