Pentagonal pyramidal number
Encyclopedia
A pentagonal pyramidal number is a figurate number
that represents the number of objects in a pyramid with a pentagonal base. The nth pentagonal pyramidal number is equal to the sum of the first n pentagonal number
s.
The first few pentagonal pyramidal numbers are:
1, 6, 18
, 40
, 75
, 126
, 196
, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 .
The formula for the nth pentagonal pyramidal number is
so the nth pentagonal pyramidal number is the average of n2 and n3. The nth pentagonal pyramidal number is also n times the nth triangular number
.
The generating function
for the pentagonal pyramidal numbers is
Figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions...
that represents the number of objects in a pyramid with a pentagonal base. The nth pentagonal pyramidal number is equal to the sum of the first n pentagonal number
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...
s.
The first few pentagonal pyramidal numbers are:
1, 6, 18
18 (number)
18 is the natural number following 17 and preceding 19.In speech, the numbers 18 and 80 are sometimes confused. When carefully enunciated, they differ in which syllable is stressed: 18 vs 80 . However, in dates such as 1864, or when contrasting numbers in the teens, such as 17, 18, 19, the stress...
, 40
40 (number)
40 is the natural number following 39 and preceding 41.Despite being related to the word "four" , 40 is spelled "forty", and not "fourty"...
, 75
75 (number)
75 is the natural number following 74 and preceding 76.-In mathematics:As the sum of the first five pentagonal numbers, seventy-five is a pentagonal pyramidal number. It is also an enneagonal number....
, 126
126 (number)
126 is the natural number following 125 and preceding 127.-In mathematics:One hundred [and] twenty-six is a pentagonal pyramidal number, and a decagonal number as well as a pentatope number....
, 196
196 (number)
196 is the natural number following 195 and preceding 197.-In mathematics:* 196 is an even number* 196 is an abundant number, as 203 is greater than 196* 196 is a composite number...
, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 .
The formula for the nth pentagonal pyramidal number is
so the nth pentagonal pyramidal number is the average of n2 and n3. The nth pentagonal pyramidal number is also n times the nth triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
.
The generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
for the pentagonal pyramidal numbers is