Some arithmetic problems raised by rabbits, cows and the Da Vinci Code

In 1202, the Italian mathematician Leonardo da Pisa, alias Fibonacci, introduced a sequence of numbers that nowadays bears his name. Under the assumption that rabbits breed (producing a pair of rabbits) when they are two months old, the Fibonacci numbers give the number of pairs of rabbits month after month after a single pair of rabbits begins breeding. According to this growth law, the number of pairs of rabbits is successively

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584 ...

In the Da Vinci Code, the first eight numbers of this sequence are used in some encrypted message. In the 14-th century, Narayana, an Indian mathematician, proposed the following problem: A cow produces one calf every year. Beginning in its fourth year, each adult calf produces one calf at the beginning of each year. How many cows and calves are there altogether after, for example, 19 years?

The sequence of integers produced by this process is now

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595 ...

These sequences of integers, related to Fibonacci's rabbits and Narayana's cows, as well as other similar sequences, raise a number of interesting arithmetic problems, some of which have been solved recently, while others are still open.