Let be a sequence that is recursively defined as , for all , with . The infinite sum, , is a fraction of the form , where and are coprime integers. What is the value of ?
The sequence satisfies
for all non-negative integers with . If , determine the value of
Define a subset of the first positive integers to be uneven if, for all , . For example, is an uneven subset, while is not. If represents the number of uneven subsets, find the remainder when is divided by .
Notes:
The empty set is considered to be an uneven subset.
You may want use a calculator at the end.
For a positive integer , consider the two recurrence relations above subjected to the conditions and .
If the value of the expression can be expressed as , where is one of the terms in the recurrence relations sequence above and and are pairwise coprime integers.
Find the value of .
For whole numbers , consider the recurrence relation defined as above with .
Find