Pentagonal number

Nested pentagons made up of 22 spheres

A pentagonal number or pentagonal number is a number that extends the concept of triangular and square numbers to the regular pentagon . However, the resulting pattern is far less symmetrical than that of the triangle and square numbers. The -th pentagonal number corresponds to the number of balls that are needed to lay a pattern with regular pentagons that have a common corner. ${\displaystyle n}$ ${\displaystyle n}$

For a figuratively even coverage see → Centered pentagonal number .

The first (off-center) pentagonal numbers are

0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ... (sequence A000326 in OEIS )

For some authors, the zero is not a pentagonal number, so the sequence of numbers only starts with the one.

The -th pentagonal number can be found with the formula ${\displaystyle n}$

${\displaystyle {\frac {n(3n-1)}{2}}}$

to calculate.

The most important statement about pentagonal numbers is the pentagonal number theorem .

Pentagonal numbers of the second kind

If you substitute for a negative whole number, you get pentagonal numbers of the second kind or also house numbers . House of cards numbers because the numbers indicate how many cards are needed to build a house of cards with floors. ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle {\frac {n(3n-1)}{2}}={\frac {m(3m+1)}{2}}}$for and ${\displaystyle m=-1\cdot n}$ ${\displaystyle n\leq 0}$

The sequence of the house numbers begins: (Sequence A005449 in OEIS ) ${\displaystyle 0,2,7,15,26,40,57,\dots }$

The house numbers can be generated as the sum of triangular numbers:

House numbers as the sum of triangular numbers

${\displaystyle 2\cdot {\frac {m(m+1)}{2}}+{\frac {(m-1)m}{2}}={\frac {m(3m+1)}{2}}}$

Web links

• House numbers