Linear Transformations of Polygonal Numbers
Problem 647
It is possible to find positive integers and such that given any triangular number, , then is always a triangular number. We define to be the sum of over all such possible pairs with . For example .
Polygonal numbers are generalisations of triangular numbers. Polygonal numbers with parameter we call -gonal numbers. The formula for the th -gonal number is where . For example when we get the formula for triangular numbers.
The statement above is true for pentagonal, heptagonal and in fact any -gonal number with odd. For example when we get the pentagonal numbers and we can find positive integers and such that given any pentagonal number, , then is always a pentagonal number. We define to be the sum of over all such possible pairs with .
Similarly we define for odd . You are given where the sum is over all odd .
Find where the sum is over all odd