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.
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.