what is the lucas number?

Answer 1

The series of numbers:
2, 1, 3, 4, 7, 11, 18, 29, 47...
is called the Lucas numbers.

It is similiar to the fibonacci seq, except you start the Lucas series with 2 & 1. You start the fibonacci series with 0 & 1.

fibonacci series:
0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

The ratio of two fibonacci number's approaches
the Phi - 1.
Phi = 1.618...Phi-1= .618
The ratio of two Lucas number's also approcahes .618... For more info see below link. The fibonacci seq is well known in Math. The Golden Ratio is 1.618 is well known in Math, Art & nature.
Phi = 1.618
1.618= 1/.618 or
.618=1/(1+.618)

Check out this link-
http://en.wikipedia.org/wiki/Lucas_number
Following quoted from Wikipedia
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START of QUOTE from Wikipedia
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"The Lucas numbers are a integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Much like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate predecessors. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio.

However, the first two Lucas numbers are L0 = 2 and L1 = 1 instead of 0 and 1, and the properties of the Lucas sequence are therefore somewhat different from those of the Fibonacci sequence. The sequence of Lucas numbers begins:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (sequence A000032 in OEIS)
The Lucas numbers are related to the Fibonacci numbers by the identities

Ln = Fn − 1 + Fn + 1,
F2n = LnFn.
Their closed formula is given as:
where is the golden ratio."
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END of QUOTE from Wikipedia
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Also check out this site:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html#Lucas
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START of QUOTE from mcs.surrey.ac.uk
on Lucas Numbers
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"The Lucas series
The French mathematician, Edouard Lucas (1842-1891), who gave the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, .. the name the Fibonacci Numbers, found a similar series occurs often when investigatng Fibonacci number patterns:
2, 1, 3, 4, 7, 11, 18, ...
The Fibonacci rule of adding the latest two to get the next is kept, but here we start from 2 and 1 (in this order) instead of 0 and 1 for the (ordinary) Fibonacci numbers.
The series, called the Lucas Numbers after him, is defined as follows: where we write its members as Ln, for Lucas:
Ln = Ln-1 + Ln-2 for n>1
L0 = 2
L1 = 1
and here are some more values of Ln together with the Fibonacci numbers for comparison:"
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END of QUOTE from mcs.surrey.ac.uk on Lucas
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Try using Google and search for Lucas and Lucas Numbers and you'll find many wonderful sites that explain this.

Sources: MS in Applied Math & MS in Math with over 100 credits toward a Phd in Math, BS in Math.

Answer 2

The Lucas numbers are a integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Much like the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate predecessors. Consequently, the ratio between two consecutive Lucas numbers converges to the golden ratio.

However, the first two Lucas numbers are L0 = 2 and L1 = 1 instead of 0 and 1, and the properties of the Lucas sequence are therefore somewhat different from those of the Fibonacci sequence. The sequence of Lucas numbers begins:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (sequence A000032 in OEIS)

Answer 3

The French mathematician, Edouard Lucas (1842-1891), who gave the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, .. the name the Fibonacci Numbers, found a similar series occurs often when investigatng Fibonacci number patterns:
2, 1, 3, 4, 7, 11, 18, ...
The Fibonacci rule of adding the latest two to get the next is kept, but here we start from 2 and 1 (in this order) instead of 0 and 1 for the (ordinary) Fibonacci numbers.

Answer 4

There is not one Lucas number, there are numbers which are part of the Lucas sequence and they are known as THE LUCAS NUMBERS. They are all integers, and they form an infinite series.