Which triangular numbers are three times a pentagonal number?
Is there a geometrical interpretation to this?
I'm not quite sure what's meant by 'geometrical interpretation'.
A figurate demonstration is possible, starting from the insight here about centered hexagonal numbers (see diagram). If a pentagonal number is constructed as demonstrated, then the required triangular number is obtained by completing the triangle - adding dots at the top and sides, plus a row at the bottom. There are then (n-1) rows of dots at the top, the original n rows, and n rows at the bottom: (n-1)+n+n=3n-1, the total number of rows and hence the index of the corresponding triangular number (by construction the result is obviously triangular).
This method doesn't even require either value to be computed, just their indices, so is actually quite neat.
Posted by broll
on 2016-07-01 10:54:34