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Count down to Christmas with maths!

Where do the magnetic fields of planets and stars come from?

- In this podcast Paul Shepherd tells us about the maths of football stadiums and why his work required him to listen to Belgian techno.
Amidst all the controversy of the FIFA World Cup 2022 there is also some football to be played. And where there's football, there's maths...

Tosin uses maths to guarantee the continued production of chocolate.

Another way we can define the Fibonacci (or 2-bonacci) sequence is A(i) = A2(i-1) - A(i-3), where i is the index or position of number A in the sequence. Any number at i in the sequence equals the one just before, i-1, multiplied by 2, minus the 3rd number back. So for example 34 = (21x2) - 8. In general an n-bonacci is defined by A(i) = A2(i-1) - A(i-(n+1)).

What if we add these two terms instead of subtracting them? What if we define an additive instead of a subtractive 3-nacci, A(i) = 2A(i-1) + A(i-4)? I got a ratio constant of about 2.1069, the positive solution to x^4 - 2x^3 - 1 = 0. In the general case of an additive n-nacci, or n-addinacci, the constant also converges on 2 but from a direction opposite to the more familiar subtractive since the 0-addinacci constant is 3 as A(i) = 2A(i-1)+ A(i-1) = 3A(i-1). The 1-addinacci (aka Pell sequence) at A(i) = 2(Ai-1) + A(Ai-2) has a constant of 2.414 (aka the Silver ratio), which is the positive solution for x^2 - 2x^1 - 1 = 0. The 2-addinacci must have the polynomial x^3 - 2x^2 - 1 = 0, and the 0-addinacci x^1 - 2x^0 - 1 = 0.

Now what about a family of sequences whose constants descend not from 3 but infinity? I suggest take the familiar Fibonacci polynomial x^2 - x- 1 = 0. The positive solution for x is here given by the fraction (1+√5)/2 ~ 1.618. Multiply just the first term of the polynomial by 2 to get the next family member 2x^2 - x - 1 = 0, and now x = (1+√9)/4 = 1. 3x^2 - x - 1 = 0 gets (1+√13)/6 ~0.768. The number under the radical increases by 4 each time, and the denominator decreases by 2. So if the first term of the polynomial is 0, as in 0x^2 + x - 1 = 0, then we should get the fraction (0 + √1)/0.

The article says anything divided by 0 is undefined, but isn't there a case for saying any natural number divided by 0 is infinity? Most of us know that the Fibonacci sequence sets out the increase in the number of rabbits that can be expected if they breed at a certain rate starting in the second month of life, in this case about 1.618 per generation, the Fibonacci constant. In rabbit breeding terms a rate of infinity means rabbits breeding more rabbits as soon as they're born, and those baby rabbits doing the same.