An -step
Fibonacci sequence is defined by letting for , , and other terms according to the linear
recurrence equation

(1)

for .

Using Brown's criterion, it can be shown that the -step
Fibonacci numbers are complete; that is, every positive number can be written as
the sum of distinct -step Fibonacci numbers. As discussed by Fraenkel (1985), every
positive number has a unique Zeckendorf-like
expansion as the sum of distinct -step Fibonacci numbers and that sum does not contain
consecutive -step
Fibonacci numbers. The Zeckendorf-like expansion can be computed using a greedy
algorithm.

The first few sequences of -step Fibonacci numbers are summarized in the table below.

The probability that no runs of consecutive tails will occur in coin tosses is given by ,
where
is a Fibonacci -step
number.

The limit
is called the -anacci
constant and given by solving

(2)

or equivalently

(3)

for
and then taking the realroot .
For even , there are exactly two real roots, one greater than 1 and
one less than 1, and for odd , there is exactly one real
root, which is always .

An exact formula for the th -anacci number can be given by