I have my own set of numbers!

Because Fibonacci numbers are quite abused in programming, a similar concept.

L0 = L1 = 1

Ln = Ln-2 + Ln-1 + 1

My first impulse is to describe them in recursive way:

def leonardo(n):
    if n in (0, 1):
        return 1
    return leonardo(n - 2) + leonardo(n - 1) + 1 

for i in range(NUMBER):
    print('leonardo[{}] = {}'.format(i, leonardo(i)))

But this is not very efficient to calculate them, as for each is calculating all the previous ones, recursively.

Here memoization works beautifully

cache = {}

def leonardo(n):
    if n in (0, 1):
        return 1

    if n not in cache:
        result = leonardo(n - 1) + leonardo(n - 2) + 1
        cache[n] = result

    return cache[n]

for i in range(NUMBER):
    print('leonardo[{}] = {}'.format(i, leonardo(i)))

Taking into account that it uses more memory, and that calculating the Nth element without calculating the previous ones is also costly.

I saw this on Programming Praxis, and I like a lot the solution proposed by Graham on the comments, using an iterator.

def leonardo_numbers():
    a, b = 1, 1
    while True:
        yield a
        a, b = b, a + b + 1

The code is really clean.