tcc[Sequence A006369, Nearest integer to 4n/3, or 2n/3

This sequence, Sloane's A006369, consists of odd numbers in pairs separated by a single even number:

0,   1, 3,   2,   5, 7,   4,   9, 11,   6,   13, 15,   8,   17, 19,   10,   21, 23,   12,   ...

Its generating function is (1+z2)(z2+3z+1)/((z-1)(z2+z+1))2. By substituting a negative power of 10 for z (such as z=1/100) we get a rational fraction whose decimal expansion gives the start of the sequence:

let z = 1/100
then (1+z2)(z2+3z+1)/((z-1)(z2+z+1))2 = 10302030100/999998000001
which is 0.010302050704091106131508171910212312...

This sequence was studied by Collatz in connection with the 3X+1 problem, but the reason I made this page for A006369 is to disprove the popular notion that "one half of the integers are even".

One Third of the Integers are Even

A006369 serves as a useful example of a false "proof" that only one third of the integers are even. By ordering the integers this way, and including the negatives on a second line:

0, 1, 3, 2, 5, 7, 4, 9, 11, 6, 13, 15, ...
-1, -3, -2, -5, -7, -4, -9, -11, -6, -13, -15, ...

we see that every integer appears exactly once, but there seem to be "twice as many" odd integers.

Of course, we could equally well have started with the sequence 0, 2, 1, 4, 6, 3, 8, 10, 5, 12, 14, 7, ... to demonstrate that one third of the integers are even!

Clearly, it is meaningless to make either claim, and therefore any claim that "half of the integers are even" is also meaningless. In an infinite set, familiar concepts like "half of the elements in the set" have no meaning.


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