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The Fibonacci Series and the Lucas SeriesEdit

The following table shows terms 0 to 20 of the Fibonacci series. With each nth term is shown the 2nth term. As the table shows, any Fibonacci term F2n is a multiple of Fn.


n

Fn

2n

F2n

F2n÷Fn

n

Fn

2n

F2n

F2n÷Fn

0

0

0

0

(2)

11

89

22

17711

199

1

1

2

1

1

12

144

24

46368

322

2

1

4

3

3

13

233

26

121393

521

3

2

6

8

4

14

377

28

317811

843

4

3

8

21

7

15

610

30

832040

1364

5

5

10

55

11

16

987

32

2178309

2207

6

8

12

144

18

17

1597

34

5702887

3571

7

13

14

377

29

18

2584

36

14930352

5778

8

21

16

987

47

19

4181

38

39088169

9349

9

34

18

2584

76

20

6765

40

102334155

15127

10

55

20

6765

123

The right hand column gives the result of dividing F2n by Fn. This produces the series (starting at n = 1), 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, ... .

This series has the same property as the Fibonacci series in that each term is the sum of the two before it. It is called the Lucas series, after Édouard Lucas (1842-1891), a French mathematician who extensively studied both series. He also invented the Tower of Hanoi puzzle.

n T(n) 2n T(2n)

T(2n)

T(n)

3n T(3n)

T(3n)

T(n)

4n T(4n)

T(4n)

T(n)

1 1 2 1 1 3 2 2 4 3 3
2 1 4 3 3 6 8 8 8 21 21
3 2 6 8 4 9 34 17 12 144 72
4 3 8 21 7 12 144 48 16 987 329
5 5 10 55 11 15 610 122 20 6765 1353
6 8 12 144 18 18 2584 323 24 46368 5796
7 13 14 377 29 21 10946 842 28 317811 24447
8 21 16 987 47 24 46368 2208 32 2178309 103729
9 34 18 2584 76 27 196418 5777 36 14930352 439128

10

55

——

20

————

6765

———

123

——

30

————

832040

———

15128

———

40

————

102334155

—————

1860621

————

The figures in the T(2n) / T(n) column show a number series with the same property as the Fibonacci: each number in the series is the sum of the two before it. However, instead of beginning 0, 1,… it begins 2, 1,…, with 2 instead of 0 as the 0th term. This series is called the Lucas series. Terms 0 to 40 of both series are shown in the table below.

——

F

——

L

————

———

F

————

L

————

—————

F

—————

L

0 0 2 14 377 843 28 317811 710647
1 1 1 15 610 1364 29 514229 1149851
2 1 3 16 987 2207 30 832040 1860498
3 2 4 17 1597 3571 31 1346269 3010349
4 3 7 18 2584 5778 32 2178309 4870847
5 5 11 19 4181 9349 33 3524578 7881196
6 8 18 20 6765 15127 34 5702887 12752043
7 13 29 21 10946 24476 35 9227465 20633239
8 21 47 22 17711 39603 36 14930352 33385282
9 34 76 23 28657 64079 37 24157817 54018521
10 55 123 24 46368 103682 38 39088169 87403803
11 89 199 25 75025 167761 39 63245986 141422324
12 144 322 26 121393 271443 40 102334155 228826127

13

233

521

27

196418

439204

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