|
gptkbp:instance_of
|
gptkb:Company
|
|
gptkbp:application
|
used in number theory.
|
|
gptkbp:closed_form
|
L(n) = (phi^n + psi^n) where phi and psi are the roots of x^2 -x -1.
|
|
gptkbp:defines
|
a sequence of numbers defined by a recurrence relation similar to the Fibonacci sequence.
|
|
gptkbp:generating_function
|
G(x) = (2 -x) / (1 -x -x^2).
|
|
gptkbp:has_property
|
they are integer sequences.
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Lucas sequences
|
|
gptkbp:l(0)
|
2
|
|
gptkbp:l(1)
|
gptkb:1
|
|
gptkbp:l(10)
|
123
|
|
gptkbp:l(2)
|
gptkb:3
|
|
gptkbp:l(3)
|
gptkb:4
|
|
gptkbp:l(4)
|
gptkb:7
|
|
gptkbp:l(5)
|
gptkb:11
|
|
gptkbp:l(6)
|
gptkb:18
|
|
gptkbp:l(7)
|
29
|
|
gptkbp:l(8)
|
47
|
|
gptkbp:l(9)
|
gptkb:76
|
|
gptkbp:l(n)
|
L(n) = F(n) + 2 F(n-1) where F is the Fibonacci sequence.
|
|
gptkbp:l(n)_for_n=0
|
2.
|
|
gptkbp:l(n)_for_n=1
|
1.
|
|
gptkbp:l(n)_for_n=10
|
123.
|
|
gptkbp:l(n)_for_n=11
|
199.
|
|
gptkbp:l(n)_for_n=12
|
322.
|
|
gptkbp:l(n)_for_n=13
|
521.
|
|
gptkbp:l(n)_for_n=14
|
843.
|
|
gptkbp:l(n)_for_n=15
|
1364.
|
|
gptkbp:l(n)_for_n=16
|
2207.
|
|
gptkbp:l(n)_for_n=17
|
3571.
|
|
gptkbp:l(n)_for_n=18
|
5778.
|
|
gptkbp:l(n)_for_n=19
|
9349.
|
|
gptkbp:l(n)_for_n=2
|
3.
|
|
gptkbp:l(n)_for_n=20
|
15127.
|
|
gptkbp:l(n)_for_n=21
|
24476.
|
|
gptkbp:l(n)_for_n=22
|
39603.
|
|
gptkbp:l(n)_for_n=23
|
64079.
|
|
gptkbp:l(n)_for_n=24
|
103682.
|
|
gptkbp:l(n)_for_n=25
|
167761.
|
|
gptkbp:l(n)_for_n=26
|
271443.
|
|
gptkbp:l(n)_for_n=27
|
439204.
|
|
gptkbp:l(n)_for_n=28
|
710647.
|
|
gptkbp:l(n)_for_n=29
|
1141851.
|
|
gptkbp:l(n)_for_n=3
|
4.
|
|
gptkbp:l(n)_for_n=30
|
1852498.
|
|
gptkbp:l(n)_for_n=31
|
2994349.
|
|
gptkbp:l(n)_for_n=32
|
4846847.
|
|
gptkbp:l(n)_for_n=33
|
7841196.
|
|
gptkbp:l(n)_for_n=34
|
12628043.
|
|
gptkbp:l(n)_for_n=35
|
20469239.
|
|
gptkbp:l(n)_for_n=36
|
33097282.
|
|
gptkbp:l(n)_for_n=37
|
53566521.
|
|
gptkbp:l(n)_for_n=38
|
86663703.
|
|
gptkbp:l(n)_for_n=39
|
140830724.
|
|
gptkbp:l(n)_for_n=4
|
7.
|
|
gptkbp:l(n)_for_n=40
|
227494427.
|
|
gptkbp:l(n)_for_n=41
|
368325151.
|
|
gptkbp:l(n)_for_n=42
|
595819578.
|
|
gptkbp:l(n)_for_n=43
|
964144729.
|
|
gptkbp:l(n)_for_n=44
|
1551968307.
|
|
gptkbp:l(n)_for_n=45
|
2516113036.
|
|
gptkbp:l(n)_for_n=46
|
4068081343.
|
|
gptkbp:l(n)_for_n=47
|
6584194379.
|
|
gptkbp:l(n)_for_n=48
|
10612215722.
|
|
gptkbp:l(n)_for_n=49
|
17196410101.
|
|
gptkbp:l(n)_for_n=5
|
11.
|
|
gptkbp:l(n)_for_n=50
|
27808625823.
|
|
gptkbp:l(n)_for_n=6
|
18.
|
|
gptkbp:l(n)_for_n=7
|
29.
|
|
gptkbp:l(n)_for_n=8
|
47.
|
|
gptkbp:l(n)_for_n=9
|
76.
|
|
gptkbp:phi
|
(1 + sqrt(5)) / 2.
|
|
gptkbp:psi
|
(1 -sqrt(5)) / 2.
|
|
gptkbp:recurrence_relation
|
L(n) = L(n-1) + L(n-2) for n > 1.
|
|
gptkbp:relation_to_fibonacci
|
Lucas numbers are related to Fibonacci numbers.
|
|
gptkbp:term_start
|
gptkb:1
2
|
|
gptkbp:bfsParent
|
gptkb:Jean-Karl_Lucas
|
|
gptkbp:bfsLayer
|
5
|