Tetrahedral Numbers which are Sum of 2 Tetrahedral Numbers
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Theorem
The sequence of tetrahedral numbers which are the sum of two other tetrahedral numbers begins:
- $20, 680, 29260, 34220, 70300, \dots$
This sequence is A034404 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
| \(\ds \) | \(\) | \(\ds 20\) | the $4$th tetrahedral number | |||||||||||
| \(\ds \) | \(=\) | \(\ds 10\) | the $3$rd tetrahedral number | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 10\) | the $3$rd tetrahedral number | ||||||||||
| \(\ds \) | \(\) | \(\ds 680\) | the $15$th tetrahedral number | |||||||||||
| \(\ds \) | \(=\) | \(\ds 120\) | the $8$th tetrahedral number | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 560\) | the $14$th tetrahedral number | ||||||||||
| \(\ds \) | \(\) | \(\ds 29 \, 260\) | the $55$th tetrahedral number | |||||||||||
| \(\ds \) | \(=\) | \(\ds 27 \, 720\) | the $54$th tetrahedral number | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 1540\) | the $20$th tetrahedral number | ||||||||||
| \(\ds \) | \(\) | \(\ds 34 \, 220\) | the $58$th tetrahedral number | |||||||||||
| \(\ds \) | \(=\) | \(\ds 29 \, 260\) | the $55$th tetrahedral number | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 4960\) | the $30$th tetrahedral number | ||||||||||
| \(\ds \) | \(\) | \(\ds 70 \, 300\) | the $74$th tetrahedral number | |||||||||||
| \(\ds \) | \(=\) | \(\ds 59 \, 640\) | the $70$th tetrahedral number | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 10 \, 660\) | the $39$th tetrahedral number |
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $680$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $680$