# Discrete correlations of order 2 of generalised Rudin-Shapiro sequences: a combinatorial approach

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the nth term of the sequence as being the total weight of the integer n written in base k. Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they have the same correlations of order 2 as sequences of symbols chosen uniformly and independently at random. The speed of convergence is very fast and is independent of the prime factor decomposition of k. This extends recent work of Tahay. The proof relies on direct observations about base-k representations of integers and combinatorial considerations. We also provide extensions of our results to higher-dimensional block-additive sequences.

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