The recursive formula for generalized pentagonal numbers characteristic function

Question

I am investigating the pentagonal number characteristic function. For the generalized pentagonal numbers the formula is known: $$p_n=n(3n\pm1)/2$$ The first few generalized pentagonal numbers are: $${0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77}$$ The characteristic function is: $${1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1}$$ I am trying to find a way to represent the characteristic function in a recursive way. Frankly saying I was not able even to find a way to start this. I have looked at the wolfram, wikipedia and oeis to find anything related to this. But nothing found.

Answer 1

Generate the generalized pentagonal number pairs: $(3r^2-r)/2$ and $(3r^2+r)/2$ starting with $r=1$.

r| pair 1| 1,2 2| 5,7 3| 12,15 and continue

If $cr$ is the characteristic function let $cr(0) = 1$. Each of the pair of pentagonal numbers are the index for the characteristic function and the sign is = $(-1)^r$ . If $cr$ is the characteristic function:

$cr(1) = cr (2) = -1$

$cr(5) = cr (7) = 1$

$cr (12) = cr (15) =-1$ etc.

cr (Non Pentagonal #s > 0) = 0