The well-known Lucas series
1, 3, 4, 7, 11, 18, 29, 47, 76, ……is closely related to the Fibonacci sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,….
The Lucas is obtained by continually adding the n and n + 2 terms of the Fibonacci to obtain the corresponding nth term of the Lucas.
Thus 1 - the 1st term of the Lucas - is obtained by adding 0 and 1 (i.e. the 1st and 3rd terms of the Fibonacci); 3 the 2nd term of the Lucas is obtained by adding 1 and 2 (i.e. the 2nd and 4th terms of the Fibonacci); 4 the 3rd term of the Lucas is obtained by adding 1 and 3 (i.e. the 3rd and 5th terms) etc.
In both the Fibonacci and Lucas the ratio of successive terms i.e. (k+ 1)/k approximates to the value of Phi (1.61803309887..).
Furthermore in the Lucas Series the nth term of the series approximates (phi)n.
Thus (phi)9 = 76.013.. which is closely approximated by the 9th term of the Lucas Series (76).
Now we can obtain a fascinating Lucas equivalent for the 3-term Fibonacci-like sequence (i.e. where starting with 0, 0, 1 we continually add the last 3terms to obtain the next term).
As we have seen this results in the sequence
0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415,.....
Now a fascinating Lucas type series can be generated for this series using the following formula
(n + 3) + 2(n + 1) + (n - 2)/2 + (n - 6)/2, where n represents the nth term of the 3-term Fibonacci series = the nth term of the corresponding Lucas series. (When the value of n is negative, it is ignored).
Thus when n = 1, we get 1 + 2(0) = 1; The other terms lead to negative values or 0, they are ignored.
Therefore the 1st term of the corresponding Lucas series is 1.
When n = 2, we get 2 + 2 (1) = 4; again the other terms are ignored.
Therefore the 2nd term of the corresponding Lucas series is 4.
When n = 3, we get 4 + 2(1) = 6; again the other terms are ignored.
Therefore the 3rd term of the corresponding Lucas series is 6.
When n = 4 we get 7 + 2 (2) = 11.
Therefore the 4th term of the corresponding Lucas series is 11.
When n = 5, we get 13 + 2(4) + 1/2 = 21.5; the final term is ignored. For convenience we can round here to 21 (to express as a whole number).
Therefore the 5th term of the corresponding Lucas sequence is 21.
When n = 6, we get 24 + 2(7) + 1/2 = 38.5; the final term is ignored. For convenience we can round down here to 38 (again to express as a whole number).
Therefore the 6th term of the corresponding Lucas sequence is 38.
When n = 7, we get 44 + 2(13) + 2/2 = 71; the final term is ignored.
Therefore the 7th term of the corresponding Lucas sequence is 71.
When n = 8, we get 81 + 2(24) + 4/2 = 131.
Therefore the 8th term of the corresponding Lucas sequence is 131.
When n = 9 we get 149 + 2(44) + 7/2 + 1/2 = 241.
Continuing on, the 10th term we get 274 + 2(81) + 13/2 + 1/2 = 443.
In like manner, the 11th term is 504 + 2(149) + 24/2 + 2/2 = 815
The 12th term is 927 + 2(274) + 44/2 + 4/2 = 1499.
The 13th term is 1705 + 2(504) + 81/2 +7/2 = 2757
The 14th term is 3136 + 2(927) + 149/2 + 13/2 = 5071
The 15th term is 5768 + 2(1705) + 274/2 + 24/2 = 9327
Therefore the first 15 terms of the corresponding Lucas series for the 3-term case are
1, 4, 6, 11, 21, 38, 71, 131, 241, 443, 815, 1499, 2757, 5071, 9327, 10609, ...
The ratio here (as in the 3-term Fibonacci) approximates to phi3= 1.839286755…For example t15/ t14 = 9327/5071 = 1.839282.. (which is correct for the first 5 figures after the decimal point).
Also (phi3)n approximates closely the value of Tn (in the Lucas series).
Thus (phi3)5 = (1.839286755..)5 = 21.0497.. This closely approximates T5 = 21.
(Phi3)10 = (1.839286755..)10 = 443.0925.. This closely approximates T10 = 443.
(Phi3)15 = (1.839286755..)15 = 9326.9930.. This closely approximates T15 = 9327.
T20 = 121415 + 2(35890) + 5768/2 + 504/2 = 196331.
(Phi3)20 = (1.839286755..)20 = 196330.99549... This closely approximates
T20 = 196331.