BBP-type formulas -- an elementary approach (Q2112783)

A BBP-type formula is a series of the form \[ \mathrm{BBP}(d,b,n,A)=\sum_{k=0}^\infty\frac{1}{b^k}\sum_{j=1}^n\frac{a_j}{(k n+j)^d} \] where \(d,b,n\in\mathbb N\) are called the degree, base and number respectively, and \(A=(a_1,\dots,a_n)\in R_n\) is a vector. It is originated in the formula by \textit{D. Bailey} et al. [Math. Comput. 66, No. 218, 903--913 (1997; Zbl 0879.11073)] as \(\mathrm{BBB}(1,16,8,A)=\pi\) with \(A=(4,0,0,-2,-1,-1,0,0)\). The purpose of this paper is to give a less general form of the main theoretical tool of \textit{D. Barsky} et al. [Acta Arith. 198, No. 4, 401--426 (2021; Zbl 1476.11103)], which as it turns out admits a simple and intuitive geometric interpretation. Aside from rediscovering some already known formulas, the method has been used in the discovery of a new BBP-type formula for \(\sqrt{3}\pi\). In addition, it is rather simple to implement an algorithm based on the main result, which is flexible enough to deal with a number of interesting cases. As an example of this, searching in various Pisot bases, we have discovered a formula for \(\pi\) in base \(1+\sqrt{3}\), along with additional formulas.