Sequence of Differences on Generalized Pentagonal Numbers
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Theorem
Recall the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$
Consider the sequence defined as $\Delta_n = GP_{n + 1} - GP_n$:
- $1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, \ldots$
This sequence is A026741 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Then:
- The values of $\Delta_n$ for odd $n$ consist of the odd numbers
- The values of $\Delta_n$ for even $n$ consist of the natural numbers.
Proof
Recall the definition of the generalised pentagonal numbers $GP_n$ for $n = 0, 1, 2, \ldots$
- $GP_n = \begin{cases} \dfrac {m \paren {3 m + 1} } 2 & : n = 2 m \\ \dfrac {m \paren {3 m - 1} } 2 & : n = 2 m - 1 \end{cases}$
for $n = 0, 1, 2, \ldots$
Hence:
\(\ds \Delta_{2 n - 1}\) | \(=\) | \(\ds GP_{2 n} - GP_{2 n - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n \paren {3 n + 1} } 2 - \dfrac {n \paren {3 n - 1} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {3 n^2 + n - 3 n^2 + n} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 n} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n\) |
which defines the sequence of natural numbers.
Then:
\(\ds \Delta_{2 n}\) | \(=\) | \(\ds GP_{2 n + 1} - GP_{2 n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds GP_{2 \paren {n + 1} - 1} - GP_{2 n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {n + 1} \paren {3 \paren {n + 1} - 1} } 2 - \dfrac {n \paren {3 n + 1} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {3 n^2 + 6 n + 3} - \paren {n + 1} - 3 n^2 - n} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {4 n + 2} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 n + 1\) |
which defines the sequence of odd numbers.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $22$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$