FIBONACCI AND EXTENSIONS (TWO TERMS)

In the well known Fibonacci sequence

0,  1,  1,  2,  3,  5,  8,  13,  21,  34,  55,  89,  144 , 233,  377,  610,  987,…. etc, the ratio of successive terms n/(n - 1) approximates to phi.

(For example the ratio of 987 and 610 which is 1.61803… approximates phi correct to 5 decimal places).

The value of phi which = (1 + square root of 5)/2, is the positive solution for x to the simple algebraic equation x2 - x - 1 = 0.

This represents a special case of the more general equation x2 - ax - b = 0 (where both a and b = 1).

So to generate the Fibonacci sequence we keep adding the last term (a) to the second last term (b) in the sequence to generate the next term. So in the above sequence 987 =  610 + 377 (i.e. a and b = 1).

However we can combine different multiples of a and b to generate different series with their own unique features.

For example the Pell Series is derived from adding 2a + b. This results in the sequence

0,  1,  2,  5,  12,  29,  70,  169,  408,  985, …

The ratio of the last two terms here n/(n - 1) is 985/408 = 2.41421… which approximates closely the value of (the square root of 2) + 1.

This in turn is the positive solution for x to the equation x2 - 2x - 1 = 0 (i.e. where a = 2 and b = 1).

Thus for example where a = 3, x will be the positive solution to the equation
x2 - 3x - 1 = 0.

i.e. (3 + square root of 13)/2 = 3.3027756...

The corresponding series is

0,  1,  3,  10,  33,  109,  360,  1189,  3927,…

The ratio of the last two terms n/(n - 1) given here is 3927/1189 = 3.302775… (which already aproximates the true value correct to six decimal places).

We can also generate unique series by varying the value of a (while keeping b constant).

Thus when a = 1 and b = 2 we obtain the following sequence

0,  1,  1,  3,  5,   11, 21,  43,  85,  171,  341,   683,  1365 ….

The ratio of the last two term n/(n - 1) given here is 1365/683 = 1.9985...

Though it does not converge so quickly (as when we vary a), this approximates the positive solution for x to the equation x2 - x - 2 = 0 i.e. (1 + square root of 9)/2 = 2.

Thus again for example when a = 3 and b = 1, the ratio of the terms n/(n - 1) will approximate the positive solution for x to the equation x2 - x - 3 = 0 i.e. (1 + square root 13)/2 = 2.302775....

The corresponding series is

0,  1,  1,  4,  7,  19,  40,  97,  217,  508,  1159,  2683,  6160,…

The ratio of the last two gives 2.29593.. which is only correct to two decimal places (due to slow convergence in this case).

We can of course combine multiples of a and b

Thus for 2a + 2b we generate the series

0,  1,  2,  6,  16,  48,  128,  352,  960,  2624,…

The ratio for n/(n - 1) will here be the positive solution for x to the equation
x2 - 2x - 2 = 0, which is 1 + square root 3 = 2.73205.. .

The ratio of the last two terms given 2624/960 = 2.7333 (which is approximating to the corrrect value).

Once again the Fibonacci sequence is a special limiting case of where we successively add unitary combinations (a and b) of the last two terms in the series starting 0, 1 to generate the next terms. So for the Fibonacci a and b = 1.

The ratio of successive terms n/(n - 1) and all cases involving multiple combinations of the two terms (a and b) can be given as the approximations to the positive solutions for x to the general quadratic equation x2 - ax - b = 0.

References

I wish to acknowledge my debt to the fascinating investigations by Mike McDermott on Fibonacci like sequences in "Knowledge and the Knower: Complexity and the Self" at  http://lightmind.com/Impermanence/Library/texts/mikem-00.html