Which triangular numbers are three times a pentagonal number?
Is there a geometrical interpretation to this?
P(n) = (3n^2-n)/2 [1]
T(n) = n/2(n+1) [2]
Say for some {T,P}, (n1)^2+(n1)=3(n2)^2-(n2).
Let (n1) = (3t-1). From [2]:
(3t-1)^2+(3t-1) = 9t^2-3t, a promising form, since 9t^2-3t = 3(3t^2-t), three times a square, less the number squared, as in [1]
In general, T(3n-1) = 3P(n). It at once follows that every pentagonal number is 1/3 a triangular number.
Edited on July 1, 2016, 10:10 am
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Posted by broll
on 2016-07-01 09:49:50 |
Observations on first part.
