In the 3-term series (starting with 0, 0, 1), when we combine the last 3 terms (i.e. a = 1, b = 1, c = 1), the following series results

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064,…
 

The ratio here is 1.839286755…

When (starting with 0, 0, 1) when we combine twice the last term, with the previous two terms (i.e. a = 2, b= 1, c = 1),

we get

0, 0, 1, 2, 5, 13, 33, 84, 214, 545, 1388, …

The ratio here is 2.5468 and is the solution to

x³ - 2x² - x - 1 = 0

When we combine the last with twice the second last and (once) the previous term (i.e. a = 1, b= 2, c = 1), we get

0, 0, 1, 1, 3, 6, 13, 28, 60, 129, 277, 595,…

The ratio here is 2.148 and is a solution to

x³ - x² - 2x - 1 = 0

When we combine the last with the second last and twice the previous term (a = 1, b = 1, c = 2), we get

0, 0, 1, 1, 2, 5, 9, 18, 37, 73, 146, 293,….

The ratio here = 2 and is solution to

x³ - x² - x - 2 = 0

In general terms when we combine k times the last with l times the second last and m times the previous term (i.e. a = k, b = l, c = m), the ratio of last and second last terms (as no. of terms approaches infinity) is positive real solution to

x³ - kx² - lx - m = 0

Thus when k = 2, l = 1, m = 3,

We get

0, 0, 1, 2, 5, 15, 41, 112, 310, 855, ….

The ratio = 2.758 (approx)

This is the solution of x³ - 2x² - x - 3 = 0
 
 
 
 

This result can then be extended generally to n terms.

Thus in the n term series when we combine s times the last with r times the second last with q times the third last ----- with k times the third with l times the second with m times the first,

the ratio of the last to the second last term (as the number of terms approaches infinity) is the positive real solution to the equation

xⁿ - sx^{n - 1} - rx^{n - 2} ,…...., - kx² - lx - m = 0

Thus in the 4-term series (starting with 0, 0, 0, 1) when s = 2, r = 1, q = 2, p = 1, we get

0, 0, 0, 1, 2, 5, 14, 38, 102, 275, 742, 2001,…

The ratio 2.697 (approx) is a solution to

x⁴ - 2x³ - x² - 2x - 1 = 0