ACMS reciprocal-sum formulas for negative generalized Fibonacci parameters

Let fn(s,t)f_n(s,t) be the generalized Fibonacci sequence, let Z+\mathbb{Z}^+ denote the positive integers, and let \lfloor\cdot\rfloor be the floor function. ACMS conjecture. If s>t1s>t\ge1, (s,t)(2,1)(s,-t)\ne(2,-1), and n,rZ+n,r\in\mathbb{Z}^+, then

(k=n1frk(s,t))1=frn(s,t)fr(n1)(s,t)1.\left\lfloor\left(\sum_{k=n}^{\infty}\frac{1}{f_{rk}(s,-t)}\right)^{-1}\right\rfloor=f_{rn}(s,-t)-f_{r(n-1)}(s,-t)-1.

If t=1t=-1 and s,n,rZ+s,n,r\in\mathbb{Z}^+, then

(k=n1frk(s,1)2)1=frn(s,1)2fr(n1)(s,1)21.\left\lfloor\left(\sum_{k=n}^{\infty}\frac{1}{f_{rk}(s,-1)^2}\right)^{-1}\right\rfloor=f_{rn}(s,-1)^2-f_{r(n-1)}(s,-1)^2-1.

These formulas are presented as the conjectured analogue for negative parameters of the preceding reciprocal-sum theorem. The supplied text gives no resolution evidence.

Sources & referencesView supporting material

Primary source

Soohyun Park, “Arithmetic properties of generalized Fibonacci sequences”, arXiv:1407.8086 (2014).

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