Which triangular numbers are three times a pentagonal number?
Is there a geometrical interpretation to this?
P(n) = (3n^2n)/2 [1]
T(n) = n/2(n+1) [2]
Say for some {T,P}, (n1)^2+(n1)=3(n2)^2(n2).
Let (n1) = (3t1). From [2]:
(3t1)^2+(3t1) = 9t^23t, a promising form, since 9t^23t = 3(3t^2t), three times a square, less the number squared, as in [1]
In general, T(3n1) = 3P(n). It at once follows that every pentagonal number is 1/3 a triangular number.
Edited on July 1, 2016, 10:10 am

Posted by broll
on 20160701 09:49:50 