Aperiodic Alternate Divergent Series of Integers

HYPERSECTIONS

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

$\displaystyle \sum_{\substack{n\geq 1 \\ j \text{ spacing} \\ k \text{ shifts}}} (-1)^{n-1} \, P_r(n) = \underbrace{0+\cdots+0}_{k\text{ shifts}} + P_r(1)+\underbrace{0+\cdots+0}_{j\text{ spacing}} - P_r(2)+\underbrace{0+\cdots+0}_{j\text{ spacing}} + P_r(3)+\underbrace{0+\cdots+0}_{j\text{ spacing}} - P_r(4)+\underbrace{0+\cdots+0}_{j\text{ spacing}}\pm\cdots = \underbrace{\dfrac{4-r}{8}}_{\text{seed}}$ (stable)

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