We will now consider sequences where a multiple of the last term is combined with the second last term to be alternately reversed with the same multiple of the second last term to be combined with the last.
The simplest case is where (starting with 0, 1) we combine twice the last term with the second last term and then alternately combine twice the second last with the last term.
This leads to the following sequence
0, 1, 2, 4, 10, 18, 46, 82, 210, 374, 958, 1706, 4370,….
Again this leads to two ratios emerging
The larger is approximated best here by 4370/1706 = 2.5615…
The smaller is approximated by 1706/958 = 1.7807..
There is a simple relationship as between these two ratios in that the larger (2.5615) = 2*1.7807. - 1.
Now the square of 2,5615... = 6.5615…
Therefore k(squared) = k + 4.
This enables us to solve the value of k from the equation k2 - k - 4 = 0 =
(square root of 17 + 1)/2
This in turn enables us to solve the smaller ratio as approximately =
(square root of 17 + 3)/4
This smaller root is the solution to the equation 2x2 - 3x - 1 = 0.
If we take the general equation ax2 - bx - c = 0, then this equation will give the solution approximating to the smaller ratio where the multiple used in these rotating sequences is a. (In this case this is 2).
Now if x is this smaller ratio then the larger ratio is ax - (b - a).
Thus in this case a = 2 and (b - a) = (3 - 2) = 1.
Therfore the larger ratio is 2 (1.7807..) - 1 - 2.5615…
In these sequences, the value of a = the (rotating) multiple used. b = 2a - 1, and c = 1, which gives the general equation that provides the solution approximating the value of the smaller ratio.
The larger ratio once again is then given by ax - (b - a).
Therefore the equation to approximate the value of the smaller ratio where the rotating multiple used is 3, is 3x2 - 5x - 1 = 0
Thus the positive value of x = (5 + square root of 37)/6 = 1.847127…
The smaller ratio will therefore approximate this value.
The larger ratio will approximate the value of 3x - (5 - 3) = 3*1.847127 - 2 = 3.541381…
Now the corresponding rotating sequence is obtained (starting with 0, 1) by combining 3 times the last term with the second last, and then alternately 3 times the second last with the last.
This results in the following series of values
0, 1, 3, 6, 21, 39, 138, 255, 903, 1668, 5907, 10911,…
The smaller ratio is approximated here by 10911/5907 = 1.84713… (which is the value for x correct to 5 decimal places!)
The larger ratio is approximated by 5907/1668 = 3.54136... (which is correct to 4 decimal places).
So we can give a perfectly general solution for the solution of ratios in all these cases where rotating multiples of terms are combined with a (single) previous term.
More complex explanations result from combining (alternating) rotating multiples of both terms.