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Create an increasing integer alternating sequence in MATLAB / Octave

I'm trying to find a way to create a number pattern like the one below

0,1,-2,3,-4,5....

Please note: it needs to go to 200000, but I will be splitting them up into groups of 2000.

I found a formula that looks like it would work on http://oeis.org/A181983, but when I create the formula in MATLAB / Octave, the numbers don't match up:

f_num= @(x) x / (1 + x)^2;
numval = f_num(1)
numval =  0.25000

Is there another way I should be doing this?

Solution

  • Method #1 - Using (-1)^x

    Just use a linear increment operator to go from 0 to 200000 and multiply the sequence by (-1)^(x+1) to allow the sign of the sequence to alternate:

    x = 0:200000;
y = ((-1).^(x+1)) .* x;

    The addition of the +1 is important so that the even positions get a positive sign while the odd positions get a negative sign.

    Method #2 - Using indexing

    Alternatively, you can declare the same array from 0 to 200000, index into every even position and negate the sign:

    x = 0:200000;
x(2:2:end) = -x(2:2:end);

    Method #3 - Using trigonometry and integers

    One more to throw into the mix. You know that for cos(x*pi), the output is -1 when x is odd and the output is 1 when x is even. We need to flip this for your case and ultimately use this alternating sequence to multiply with the same array going from 0 to 200000, and therefore:

    x = 0:200000;
y = (-cos(x*pi)).*x;

    Aside

    Interestingly enough, (-1)^x is also equal to exp(i*pi*x) for all values of x that are integer. We can verify this by using Euler's formula where: exp(i*pi*x) = cos(pi*x) + i*sin(pi*x). Since i*sin(pi*x) = 0 for all x belonging to an integer, we really get exp(i*pi*x) = cos(pi*x). Substituting even numbers of x will give us 1 while odd numbers of x will give us -1, and hence exp(i*pi*x) = cos(pi*x) = (-1)^x for all x belonging to integers.

    Also, (-1)^(x+1) = -(-1)^x = -cos(x*pi) for all x belonging to integers and so the first method is really equal to the third method anyway!