Aperiodic Monotonic Divergent Series of Integers

HYPERSECTIONS

j-spaced and k-shifted monotonic divergent series of polygonal numbers of order r (starting from 1)

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} P_r(n) = \underbrace{0+\cdots+0}_{k\text{ shifts}} + P_r(1)+\underbrace{0+\cdots+0}_{j\text{ spacing}} + P_r(2)+\underbrace{0+\cdots+0}_{j\text{ spacing}} + \\ P_r(3)+\underbrace{0+\cdots+0}_{j\text{ spacing}} + P_r(4)+\underbrace{0+\cdots+0}_{j\text{ spacing}}+\cdots = \underbrace{\dfrac{r-4}{24}}_{\text{seed}} \,-\, \dfrac{5r-16}{12} \left(\dfrac{k-j}{j+1}\right)^2$ (unstable)

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