POLYGONAL NUMBERS

Pn is given by the formula _____________.

(3n² - n)/2

What are triangular numbers?

A triangular number or triangle number counts the objects that can form an equilateral triangle.

What is the sequence of generalized pentagonal numbers?

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335

What is a pentagonal number?

A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex.

In the limit, the ratio between the, dots and the number of line segments between closest pairs of dots in the triangle is:

lim Tn/Ln = 1/3 n→∞

The expression for the nth square number is ______.

n²

The difference between any perfect square and its predecessor is given by the identity _________.

n² - (n - 1)² = 2n - 1

What is a formula that all triangular square numbers fit?

t(t - 1)/ 2 = s² and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation x² - 2y = 1

N² Is also equal to the sum of the ____________.

First n odd numbers. Example: 5² = 25 = 1 + 3 + 5 + 7 + 9

What are generalized pentagonal numbers?

Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, -1, 2, -2, 3, -3, 4....

What are polygonal numbers?

In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.

What is a square number?

In mathematics, a square number or perfect square is an integer that is the square of an integer;[1] in other words, it is the product of some integer with itself. The number m is a square number if and only if one can compose a square of m equal squares:

What is a square triangular number?

In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are an infinite number of square triangular numbers.

What are the first few square triangular numbers?

The first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025

What is the relationship between pentagonal and triangular numbers?

The nth pentagonal number is one third of the 3n-1th triangular number.

How do we find the nth triangular number?

The nth triangular number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n.

What is the sequence of triangular numbers?

The sequence of triangular numbers (sequence A000217 in OEIS), starting at the 0th triangular number, is: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406 ...

What is one relationship between triangular numbers and square numbers?

The sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum).

How do triangular numbers solve the handshake problem?

The triangular number Tn solves the "handshake problem" of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of n people is Tn−1.