Proof of Kimberling's ``even second column'' conjecture (Q2894227)

The author proves \textit{C. Kimberling}'s [Fibonacci Q. 32, No.4, 301--315 (1994; Zbl 0813.11010)] conjecture that in the second column of the so-called even second column array (which is an example of a Stolarsky interspersion, cf. [ibid.]) every number is even. The proof used the quasi-Zeckendorf representation of a positive integer (contrary to the standard Zeckendorf representation also Fibonacci numbers \(F_0\) and \(F_1\) are allowed to be used) introduced by the author. For instance, in this representation every positive integer has a unique representation with odd least index (and similarly for even one). The approach via quasi-Zeckendorf representation allows the author to give also alternative proofs for some known results about Stolarsky interspersion, e.g. for the so-called ``even first column'' array, or the Wythoff array.