On the Constant in the Average Digit Sum for a Recurrence-Based Numeration

Given an integral, increasing, linear-recurrent sequence A with initial term 1, the greedy algorithm may be used on the terms of A to represent all positive integers. For large classes of recurrences, the average digit sum is known to equal c_A log n+O(1), where c_A is a positive constant that depends on A. This asymptotic result is re-proved with an elementary approach for a class of special recurrences larger than, or distinct from, that of former papers. The focus is on the constants c_A for which, among other items, explicit formulas are provided and minimal values are found, or conjectured, for all special recurrences up to a certain order.