## What is the formula of polygonal numbers?

Theorem 4.1 A polygonal number with d sides and of order n is generated by the formula pd(n) = (d − 2)n2 + (4 − d)n 2 .

**What type of polygonal numbers does the numbers 1 3 6 10 15 illustrate?**

Every hexagonal number is also a triangular number 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66.

### What is the next pentagonal number?

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187… (sequence A000326 in the OEIS).

**What are the first four Heptagonal numbers?**

The first few heptagonal numbers are: 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … (sequence A000566 in the OEIS)

#### What are triangular numbers?

A triangular number is a number that can be represented by a pattern of dots arranged in an equilateral triangle with the same number of dots on each side. The first triangular number is 1, the second is 3, the third is 6, the fourth 10, the fifth 15, and so on.

**What is the formula for pentagonal numbers?**

Each pentagonal number is split into a rectangular array and a triangular number, a subdivision that suggests that we can represent the nth pentagonal number by (n-1)n + Tn, where Tn is the nth triangular number. So the sub-pattern suggests that the nth pentagonal number can be expressed as Pn = (n-1)n + n(n+1)/2.

## What are the first four pentagonal numbers?

i.e. the first pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145.

**What are the first 10 Fibonacci numbers?**

The First 10 Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.

### What is the sixth hexagonal number?

1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946… Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number.

**Why is 28 the perfect number?**

A number is perfect if all of its factors, including 1 but excluding itself, perfectly add up to the number you began with. 6, for example, is perfect, because its factors — 3, 2, and 1 — all sum up to 6. 28 is perfect too: 14, 7, 4, 2, and 1 add up to 28.

#### What are the values of a polygonal number?

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, The first 6 values in the column “sum of reciprocals”, for triangular to octagonal numbers, come from a published solution to the general problem, which also gives a general formula for any number of sides, in terms of the digamma function.

**What do you call a polygonal number with 3 sides?**

We will use the common notation p3(r) to mean the r th polygonal number with 3 sides. r is called the rank of the triangular number meaning the length of (the number of dots in) the outside edges. 3.2.5 You Do The Maths… How many dots are there on an ordinary cubical dice?

## What is the total number of dots in a set of regular polygons?

The total number PNnof dots in a set of n nested regular polygons with N sides is called a “polygonal number.” The number 1 is the first polygonal number, i.e., PN1= 1. The second polygonal number, which is simplythe number of corners of an N-sided, regular polygon is PN2= N, the third is PN3, etc.

**Are there any polygonal numbers that are m gonal?**

1.1Nontrivial polygonal numbers 1.1.1Composite numbers nwhich are not m-gonal numbers for 3 ≤ m< n 1.1.1.1Even composite numbers nwhich are not m-gonal numbers for 3 ≤ m< n 1.1.1.2Odd composite numbers nwhich are not m-gonal numbers for 3 ≤ m< n