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Connell sequence

The Connell sequence is the sequence obtained by starting with the first positive odd number (1), taking the next two even numbers (2, 4), the next three odd numbers (5, 7, 9), the next four even numbers (10, 12, 14, 16), and so on. The first few terms are 1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, ... (OEIS A001614). A binary plot of the sequence from 1 to 255 is illustrated above.Amazingly, the terms of this sequence have the closed formThis shows immediately that

Odd number

An odd number is an integer of the form , where is an integer. The odd numbers are therefore ..., , , 1, 3, 5, 7, ... (OEIS A005408), which are also the gnomonic numbers. Integers which are not odd are called even.Odd numbers leave a remainder of 1 when divided by two, i.e., the congruence holds for odd . The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.The generating function for the odd numbersisThe product of an even number and an odd number isalways even, as can be seen by writingwhich is divisible by 2 and hence is even.

Even number

An even number is an integer of the form , where is an integer. The even numbers are therefore ..., , , 0, 2, 4, 6, 8, 10, ... (OEIS A005843). Since the even numbers are integrally divisible by two, the congruence holds for even . An even number for which also holds is called a singly even number, while an even number for which is called a doubly even number. An integer which is not even is called an odd number.The oddness of a number is called its parity, so an odd number has parity 1, while an even number has parity 0.The generating function of the even numbersisThe product of an even number and an odd number isalways even, as can be seen by writingwhich is divisible by 2 and hence is even.

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