Answer:
Each term is a sum of the two previous terms like the fibonacci sequence, but the sum of the two starting terms of the lucas sequence is followed by 2 then 1, rather than 0 then 1 for the fibannaci sequence.
These are integer sequences.
The sum of the two previous terms can be represented by:
fn = fn-1 + fn-2.
These numbers can be obtained by adding the previous corresponding term of the fibonacci sequence by the next corresponding term to get the corresponding number of the lucas sequence.
Ln = { 2 if n = 0
1 if n = 1
Ln-1 + Ln-2 if n > 1
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Fn+1 + Fn-1 = Ln
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f(0) = -1
f(1) = 2, f(2) = 1, f(3) = 3, f(4) = 4, f(5) = 7, f(6) = 11
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f(12) = 199
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f(12) = f(11) + f(10) = (f(10) + f(9)) + (f(9) + f(8)) = .....
given f(n) = f(n-1) + f(n-2)
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f(12) = f(1) + f(2) = f(2) + f(3) = f(3) + f(4) =
f(4) + f(5) = f(5) + f(6) = f(6) + f(7) =
f(7) + f(8) = f(8) + f(9) = f(9) + f(10) =
f(10) + f(11) = f(12) →
f(12) = 2 + 1 = 1 + [3] = 3 + [4] = 4 + [7] =
7 + [11] = 11 + [18] = 18 + [29] = 29 + [47] =
47 + [76] = 76 + [123] = 199
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Explicit formula
for n is an integer.
Nth term of the lucas sequence.
As you can see, the golden ratio
( 1 + √5 / 2) appears in here.
L(n) = ((1 + √5) / 2)ⁿ⁻¹ + ((1 – √5) / 2)ⁿ⁻¹