Generalized Mersenne Sequence continuation

[carpetpool]

Given the Mersenne Sequence 2^n-1, it can be modeled using the true equation:

2 - 1 = 1

One generalization includes fixing 2 to a and forming the equation and sequence:

a - 1 = b

(a^n-1)/b

Another generalization include:

a - b = c

(a^n-b^n)/c

What is the proper Generalized Mersenne Sequence(s) for the "base" equation:

x*a - y*b = z*c

in terms of (a, b, c, x, y, z) and the exponent n?

In the case of (a^n-b^n)/c, (x, y, z) = 1. What about when one, both, or all of (x, y, z) > 1? What is the closed form similar to (a^n-b^n)/c with the same sequence properties?

[science_man_88]

Originally Posted by carpetpool

Given the Mersenne Sequence 2^n-1, it can be modeled using the true equation:

2 - 1 = 1

One generalization includes fixing 2 to a and forming the equation and sequence:

a - 1 = b

(a^n-1)/b

Another generalization include:

a - b = c

(a^n-b^n)/c

What is the proper Generalized Mersenne Sequence(s) for the "base" equation:

x*a - y*b = z*c

in terms of (a, b, c, x, y, z) and the exponent n?

In the case of (a^n-b^n)/c, (x, y, z) = 1. What about when one, both, or all of (x, y, z) > 1? What is the closed form similar to (a^n-b^n)/c with the same sequence properties?

https://en.wikipedia.org/wiki/Mersen...eneralizations shows complex number generalizations as well.