Closed Form for Square Pyramidal Numbers

Theorem

The closed-form expression for the $n$th square pyramidal number is:

$S_n = \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6$

Proof

\(\ds S_n\)

\(=\)

\(\ds \sum_{k \mathop = 1}^n k^2\)

Definition of Square Pyramidal Number

\(\ds \)

\(=\)

\(\ds \dfrac {n \paren {n + 1} \paren {2 n + 1} } 6\)

Sum of Sequence of Squares

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $55$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $55$