The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.

The variant where "n divides a(n) + a(n+1)" corresponds to A005408 (the odd numbers).

Empirically, the scatterplot of the first terms shows a bundle of curves that correspond to terms sharing the same value of Sum_{k=2.. n} ((-1)^k (a(k+1) + a(k))/k) (see Links section); the sequence A134204 has similar features.