A003294

A003294 is an entry in the OEIS (Online Encyclopedia of Integer Sequences), a comprehensive database of integer sequences. The sequence is defined as the number of partitions of n into parts, each of which is a power of 2. In other words, it counts the number of ways to express n as a sum of 2^a1 + 2^a2 + ... + 2^ak, where a1, a2, ..., ak are non-negative integers.

The sequence starts with 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 8, 4, 9, 1, 11, 5, 13, 3, ... (sequence A003294 in OEIS). The first few terms are:

* a(0) = 1 (the empty partition)

* a(1) = 1 (1 = 2^0)

* a(2) = 2 (2 = 2^1 or 2 = 2^0 + 2^0)

* a(3) = 1 (3 cannot be expressed as a sum of powers of 2)

* a(4) = 3 (4 = 2^2 or 4 = 2^1 + 2^1 or 4 = 2^0 + 2^0 + 2^0 + 2^0)

The sequence is related to the binary representation of numbers and has applications in computer science and combinatorics. It is also connected to other sequences in the OEIS, such as A000009 (the number of partitions of n) and A000010 (the number of partitions of n into distinct parts).

The generating function for A003294 is given by the product of (1 + x^(2^k)) for k = 0, 1, 2, ..., which reflects the fact that each term in the partition can be any power of 2. The sequence is also known as the "binary partition function" or the "2-adic partition function."

The sequence A003294 was first noted by J. H. Conway in 1973 and has since been studied by many mathematicians. It is an example of a sequence that can be defined in terms of a simple combinatorial problem but has deep connections to other areas of mathematics.