What are Pentadecagon numbers

Last time we met Tetradecagon numbers.

Today we are meeting Pentadecagon numbers.

Tetradecagon has 14 sides and Pentadecagon has 15 sides.

We can use dots on the sides to show a polygon.

Then we can build the next bigger one.

We count the dots and that is a Pentadecagon number.

Pentadecagon number

Pd(n) = n (13n — 11)/2 (1)

1 , 15 , 42 , 82 , 135 , 201 , 280 , 372 , 477 , 595 , …

We build by adding to the bottom.

From n to (n + 1) we add (13n + 1)

Use Math induction

Pd(1) = 1

Pd(n+1) = Pd(n) + 13n + 1 = n (13n — 11)/2 + 13n + 1

= (13 n² — 11n+ 26n + 2)/2

= (n+1)(13n + 2)/2= (n+1)(13(n + 1) — 11)/2

By induction eqn(1) holds.

Please enjoy and have fun! Thank you!