  For an arbitrary homogeneous linear recurrence sequence of order d with
constant coefficients, we derive recurrence relations for all subsequences with
indices in arithmetic progression. The coefficients of these recurrences are
given explicitly in terms of partial Bell polynomials that depend on at most
d-1 terms of the generalized Lucas sequence associated with the given
recurrence. We also provide an elegant formula for the partial sums of such
sequences and illustrate all of our results with examples of various orders,
including common generalizations of the Fibonacci numbers.
