which is known as Euler's pentagonal number theorem, and also (3) and
g-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue's identity and Euler's pentagonal number theorem.
That is if we suppose the pentagonal number 5, then definitely it has a structure but when it is considered as a graph then the structure gives the graphical partition of 10.
Some properties of pentagonal numbers and the relationship of partition functions of numbers have also been discussed.
It is well known that a positive integer N is called a Pentagonal Number if N = m(3m-1)/2 for any integer m > 0, and a positive integer N is called a Generalized Pentagonal Number if N = m(3m-1)/2 for any integer m.
1)] is Generalized Pentagonal Number if and only if n [equivalent to] 0, 1 or 3 (ii) [Q.
Some third of the material is concerned with biographical and other contextual issues, while the bulk of the selections focus on particular aspects of Euler's contributions to mathematics, including infinite series, the zeta functions, Euler's constant, differentials, multiple integrals, the calculus of variations, the pentagonal number
theorem, quadratic reciprocity, and the fundamental theorem of algebra.
The first pentagonal number
P(1) is 1 and the second pentagonal number
P(2) is clearly 5.
So, how do you get from one pentagonal number
to the next one?
Now your students have the background to try the next step, which involves the pentagonal numbers
(N = 5).
Muktibodh  working on the same lines has defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of pentagonal numbers
of dimension 2 using the Maple system.
Working on the same lines we have defined and investigated consecutive, reversed, mirror and symmetric Smarandache sequences of pentagonal numbers
of dimension 2 using the Maple system .