Main Index Number Theory Sequences Recurrent sequences Linear recurrent sequences Binary recurrent sequences Lucas’ sequences Lucas’ Numbers
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Lucas’ Numbers

If we take the recurrent rule typeset structure and the initial values  typeset structure, or typeset structureand typeset structure then the elements of the resulting sequences are called the Lucas numbers typeset structure. The initial segment of the sequence is

2 , 1 , 3 , 4 , 7 , 11 , 18 , 29 , 47 , 76 , 123 , 199 , 322 , 521 , 843 , 1364 , ...

To compute typeset structure for other values of typeset structure go to .

Lucas numbers can be defined also in the form

(L(n + 1)/L(n)) = (1 1)^n (1/2) ,      n >= 0. 1 0

The Binet’s formula   for Lucas numbers is .

L(n) = ((1 + 5^(1/2))/2)^n + ((1 - 5^(1/2))/2)^n .

There follows from this relation that typeset structure is the nearest integer to typeset structure. Consequently typeset structure has approximately typeset structure decimal digits (typeset structure). Similarly we get that

lim _ (n -> ∞) L(n + k)/L(n) = ((1 + 5^(1/2))/2)^k .

The generating series for Lucas numbers is

L(0) + L(1) z + L(2) z^2 + ··· = Underscript[∑, n >= 0] L(n) z^n = (2 - z)/(1 - z - z^2)

If typeset structure is the typeset structureth Fibonacci number then

L(n) = F(n - 1) + F(n + 1)(1)
F(n) = 1/5 (L(n - 1) + L(n + 1))(2)
F(n + 1) = (F(n) + L(n))/2(3)
L(n + 1) = (5 F(n) + L(n))/2 .(4)

The so-called doubling formulas say

F(2 n) = F(n) L(n)(5)
L(2 n) = L^2(n) - 2 (-1)^n .(6)

Since typeset structure, we  can extend the Lucas numbers to negative integers by typeset structure

Cite this web-page as:

Štefan Porubský: Lucas’ Numbers.

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