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 Michelle E. Chung
* EMAT6680 Assignment 12: Fibonnaci Sequence

Lucas Sequence


4. Generate a Fibonnaci Sequence in the first column using F(0)=1, F(1)=1,

f(n) = f(n-1) + f(n-2).
a. Construct the ratio of each pair of adjacent terms in the Fibonnaci sequence.
     What happens as n increases?
     What about the ratio of eery second term? etc. 

b. Explore sequences where f(0) and f(1) are some arbitary integers other than 1.
     If f(0)=1 and f(1)=3, then your sequence is a Lucas Sequence.
     All such sequences, however, have the same limit of the ratio of successive terms.

a. Fibonnaci Sequence

 

Fibonnaci Sequence Table
w/its ratios

 

    
n
Fibonnaci Sequence
f(n)/f(n-1)
f(n)/f(n-2)
f(n)/f(n-3)
f(n)/f(n-4)
1
 1      
2
 1 1    
3
 2 2 2    
4
 3 1.5 3 3  
5
 5 1.666666667 2.5 5 5
6
 8 1.6 2.666666667 4 8
7
 13 1.625 2.6 4.333333333 6.5
8
 21 1.615384615 2.625 4.2 7
9
 34 1.619047619 2.615384615 4.25 6.8
10
 55 1.617647059 2.619047619 4.230769231 6.875
11
 89 1.618181818 2.617647059 4.238095238 6.846153846
12
 144 1.617977528 2.618181818 4.235294118 6.857142857
13
 233 1.618055556 2.617977528 4.236363636 6.852941176
14
 377 1.618025751 2.618055556 4.235955056 6.854545455
15
 610 1.618037135 2.618025751 4.236111111 6.853932584
16
 987 1.618032787 2.618037135 4.236051502 6.854166667
17
 1597 1.618034448 2.618032787 4.236074271 6.854077253
18
 2584 1.618033813 2.618034448 4.236065574 6.854111406
19
 4181 1.618034056 2.618033813 4.236068896 6.854098361
20
 6765 1.618033963 2.618034056 4.236067627 6.854103343
21
 10946 1.618033999 2.618033963 4.236068111 6.85410144
22
 17711 1.618033985 2.618033999 4.236067926 6.854102167
23
 28657 1.61803399 2.618033985 4.236067997 6.85410189
24
 46368 1.618033988 2.61803399 4.23606797 6.854101996
25
 75025 1.618033989 2.618033988 4.23606798 6.854101955
26
 121393 1.618033989 2.618033989 4.236067976 6.854101971
27
 196418 1.618033989 2.618033989 4.236067978 6.854101965
28
 317811 1.618033989 2.618033989 4.236067977 6.854101967
29
 514229 1.618033989 2.618033989 4.236067978 6.854101966
30
 832040 1.618033989 2.618033989 4.236067977 6.854101966
31
 1346269 1.618033989 2.618033989 4.236067978 6.854101966
32
 2178309 1.618033989 2.618033989 4.236067977 6.854101966
33
 3524578 1.618033989 2.618033989 4.236067978 6.854101966
34
 5702887 1.618033989 2.618033989 4.236067977 6.854101966
35
 9227465 1.618033989 2.618033989 4.236067977 6.854101966
 
Golden Ratio
Square
of Golden Ratio
Cubic
of Golden Ratio

4th power
of Golden Ratio

 
Scatter Plot
of the Ratios		  

Ratio1
: f(n)/f(n-1)

     Ratio1

  • The first ratio, the ratio of each pair of adjecent terms in the Fibonnaci Sequence, is approaching 1.618033989, which is known as Golden Ratio.
Ratio2
: f(n)/f(n-2)		       Ratio2
  • The second ratio, the ratio of every second term in the Fibonnaci Sequence, is approaching 2.618033989, which is the square of Golden Ratio.
Ratio3
: f(n)/f(n-3)		       Ratio3
  • The third ratio, the ratio of every third term in the Fibonnaci Sequence, is approaching 4.236067977, which is the cubic of Golden Ratio.
Ratio4
: f(n)/f(n-4)		       Ratio4
  • The forth ratio, the ratio of every forth term in the Fibonnaci Sequence, is approaching 6.854101966, which is the 4th power of Golden Ratio.

Conclusion

  
  • As n increases, the ratios are bouncing up and down;
    however, they are approaching limits of the sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

So, we can predict, if we calculate f(n)/f(n-5), it would be limited to 5th power of Golden Ratio, and so on.

Let's check it.

Table
Scatter Plot
n
f(n)/f(n-5)
1
2
3
4
5
6
 8
7
 13
8
 10.5
9
 11.33333333
10
 11
11
 11.125
12
 11.07692308
13
 11.0952381
14
 11.08823529
15
 11.09090909
16
 11.08988764
17
 11.09027778
18
 11.09012876
19
 11.09018568
20
 11.09016393
21
 11.09017224
22
 11.09016907
23
 11.09017028
24
 11.09016982
25
 11.09016999
26
 11.09016993
27
 11.09016995
28
 11.09016994
29
 11.09016994
30
 11.09016994
31
 11.09016994
32
 11.09016994
33
 11.09016994
34
 11.09016994
35
 11.09016994
 
5th power
 of Golden Ratio
Ratio5

From the table, we can assert our conjecture.

  • The first terms of the ratios are same as Fibonnaci Sequence.
    So, if we take the first term of each ratio and make a sequence, it would be Fibonnaci Sequence.

Back to the Top

b. Lucas Sequence

 

Lucas Sequence Table
w/its ratios

 

    
n
Lucas Sequence
f(n)/f(n-1)
f(n)/f(n-2)
f(n)/f(n-3)
f(n)/f(n-4)
1
 1      
2
 3 3    
3
 4 1.333333333 4    
4
 7 1.75 2.333333333 7  
5
 11 1.571428571 2.75 3.666666667 11
6
 18 1.636363636 2.571428571 4.5 6
7
 29 1.611111111 2.636363636 4.142857143 7.25
8
 47 1.620689655 2.611111111 4.272727273 6.714285714
9
 76 1.617021277 2.620689655 4.222222222 6.909090909
10
 123 1.618421053 2.617021277 4.24137931 6.833333333
11
 199 1.617886179 2.618421053 4.234042553 6.862068966
12
 322 1.618090452 2.617886179 4.236842105 6.85106383
13
 521 1.618012422 2.618090452 4.235772358 6.855263158
14
 843 1.618042226 2.618012422 4.236180905 6.853658537
15
 1364 1.618030842 2.618042226 4.236024845 6.854271357
16
 2207 1.618035191 2.618030842 4.236084453 6.854037267
17
 3571 1.61803353 2.618035191 4.236061684 6.854126679
18
 5778 1.618034164 2.61803353 4.236070381 6.854092527
19
 9349 1.618033922 2.618034164 4.236067059 6.854105572
20
 15127 1.618034014 2.618033922 4.236068328 6.854100589
21
 24476 1.618033979 2.618034014 4.236067844 6.854102492
22
 39603 1.618033992 2.618033979 4.236068029 6.854101765
23
 64079 1.618033987 2.618033992 4.236067958 6.854102043
24
 103682 1.618033989 2.618033987 4.236067985 6.854101937
25
 167761 1.618033989 2.618033989 4.236067975 6.854101977
26
 271443 1.618033989 2.618033989 4.236067979 6.854101962
27
 439204 1.618033989 2.618033989 4.236067977 6.854101968
28
 710647 1.618033989 2.618033989 4.236067978 6.854101966
29
 1149851 1.618033989 2.618033989 4.236067977 6.854101966
30
 1860498 1.618033989 2.618033989 4.236067978 6.854101966
31
 3010349 1.618033989 2.618033989 4.236067977 6.854101966
32
 4870847 1.618033989 2.618033989 4.236067978 6.854101966
33
 7881196 1.618033989 2.618033989 4.236067977 6.854101966
34
 12752043 1.618033989 2.618033989 4.236067978 6.854101966
35
 20633239 1.618033989 2.618033989 4.236067977 6.854101966
 
Golden Ratio
Square
of Golden Ratio
Cubic
of Golden Ratio

4th power
of Golden Ratio

 
Scatter Plot
of the Ratios		  

Ratio1
: f(n)/f(n-1)

     L_Ratio1

  • The first ratio, the ratio of each pair of adjecent terms in the Lucas Sequence, starts differnt number, 3; however, it is still approaching 1.618033989, which is known as Golden Ratio, just like Fibonnaci Sequence.
Ratio2
: f(n)/f(n-2)		       L_Ratio2
  • The second ratio, the ratio of every second term in the Lucas Sequence, starts differnt number, 4; however, it is still approaching 2.618033989, which is square of Golden Ratio, just like Fibonnaci Sequence.
Ratio3
: f(n)/f(n-3)		       L_Ratio3
  • The third ratio, the ratio of every third term in the Lucas Sequence, starts differnt number, 7; however, it is still approaching 4.236067977, which is cubic of Golden Ratio, just like Fibonnaci Sequence.
Ratio4
: f(n)/f(n-4)		       L_Ratio4
  • The forth ratio, the ratio of every forth term in the Lucas Sequence, starts differnt number, 11; however, it is still approaching 6.854101966, which is 4th power of Golden Ratio, just like Fibonnaci Sequence.

Conclusion

  
  • The Lucas Sequence has different numbers from Fibonnaci Sequence. So, the ratios start from different numbers;
    however, they are approaching same limits of the sequence as Fibonnaci Sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

So, we still can predict that if we calculate f(n)/f(n-5), it would be limited to 5th power of Golden Ratio, and so on, just like Fibonnaci Sequence.

  • The first terms of the ratios are same as Lucas Sequence.
    So, if we take the first term of each ratio and make a sequence, it would be Lucas Sequence.

Back to the Top

c. Other Sequences

 

When
f(0)=1 and f(1)=10

 

    
n
Sequence
f(n)/f(n-1)
f(n)/f(n-2)
f(n)/f(n-3)
f(n)/f(n-4)
1
 1      
2
 10 10    
3
 11 1.1 11    
4
 21 1.909090909 2.1 21  
5
 32 1.523809524 2.909090909 3.2 32
6
 53 1.65625 2.523809524 4.818181818 5.3
7
 85 1.603773585 2.65625 4.047619048 7.727272727
8
 138 1.623529412 2.603773585 4.3125 6.571428571
9
 223 1.615942029 2.623529412 4.20754717 6.96875
10
 361 1.618834081 2.615942029 4.247058824 6.811320755
11
 584 1.617728532 2.618834081 4.231884058 6.870588235
12
 945 1.618150685 2.617728532 4.237668161 6.847826087
13
 1529 1.617989418 2.618150685 4.235457064 6.856502242
14
 2474 1.618051014 2.617989418 4.23630137 6.853185596
15
 4003 1.618027486 2.618051014 4.235978836 6.854452055
16
 6477 1.618036473 2.618027486 4.236102027 6.853968254
17
 10480 1.61803304 2.618036473 4.236054972 6.854153041
18
 16957 1.618034351 2.61803304 4.236072945 6.854082458
19
 27437 1.61803385 2.618034351 4.23606608 6.854109418
20
 44394 1.618034042 2.61803385 4.236068702 6.85409912
21
 71831 1.618033969 2.618034042 4.236067701 6.854103053
22
 116225 1.618033996 2.618033969 4.236068083 6.854101551
23
 188056 1.618033986 2.618033996 4.236067937 6.854102125
24
 304281 1.61803399 2.618033986 4.236067993 6.854101906
25
 492337 1.618033988 2.61803399 4.236067972 6.854101989
26
 796618 1.618033989 2.618033988 4.23606798 6.854101957
27
 1288955 1.618033989 2.618033989 4.236067977 6.85410197
28
 2085573 1.618033989 2.618033989 4.236067978 6.854101965
29
 3374528 1.618033989 2.618033989 4.236067977 6.854101967
30
 5460101 1.618033989 2.618033989 4.236067978 6.854101966
31
 8834629 1.618033989 2.618033989 4.236067977 6.854101966
32
 14294730 1.618033989 2.618033989 4.236067978 6.854101966
33
 23129359 1.618033989 2.618033989 4.236067977 6.854101966
34
 37424089 1.618033989 2.618033989 4.236067978 6.854101966
35
 60553448 1.618033989 2.618033989 4.236067977 6.854101966
 
Golden Ratio
Square
of Golden Ratio
Cubic
of Golden Ratio

4th power
of Golden Ratio

 
When
f(0)=10 and f(1)=17		  
n
Sequence
f(n)/f(n-1)
f(n)/f(n-2)
f(n)/f(n-3)
f(n)/f(n-4)
1
 10      
2
 17 1.7    
3
 27 1.588235294 2.7    
4
 44 1.62962963 2.588235294 4.4  
5
 71 1.613636364 2.62962963 4.176470588 7.1
6
 115 1.61971831 2.613636364 4.259259259 6.764705882
7
 186 1.617391304 2.61971831 4.227272727 6.888888889
8
 301 1.61827957 2.617391304 4.23943662 6.840909091
9
 487 1.617940199 2.61827957 4.234782609 6.85915493
10
 788 1.618069815 2.617940199 4.23655914 6.852173913
11
 1275 1.618020305 2.618069815 4.235880399 6.85483871
12
 2063 1.618039216 2.618020305 4.23613963 6.853820598
13
 3338 1.618031992 2.618039216 4.236040609 6.854209446
14
 5401 1.618034751 2.618031992 4.236078431 6.854060914
15
 8739 1.618033697 2.618034751 4.236063984 6.854117647
16
 14140 1.6180341 2.618033697 4.236069503 6.854095977
17
 22879 1.618033946 2.6180341 4.236067395 6.854104254
18
 37019 1.618034005 2.618033946 4.2360682 6.854101092
19
 59898 1.618033983 2.618034005 4.236067893 6.8541023
20
 96917 1.618033991 2.618033983 4.23606801 6.854101839
21
 156815 1.618033988 2.618033991 4.236067965 6.854102015
22
 253732 1.618033989 2.618033988 4.236067982 6.854101948
23
 410547 1.618033989 2.618033989 4.236067976 6.854101973
24
 664279 1.618033989 2.618033989 4.236067978 6.854101964
25
 1074826 1.618033989 2.618033989 4.236067977 6.854101967
26
 1739105 1.618033989 2.618033989 4.236067978 6.854101966
27
 2813931 1.618033989 2.618033989 4.236067977 6.854101966
28
 4553036 1.618033989 2.618033989 4.236067978 6.854101966
29
 7366967 1.618033989 2.618033989 4.236067977 6.854101966
30
 11920003 1.618033989 2.618033989 4.236067978 6.854101966
31
 19286970 1.618033989 2.618033989 4.236067977 6.854101966
32
 31206973 1.618033989 2.618033989 4.236067978 6.854101966
33
 50493943 1.618033989 2.618033989 4.236067977 6.854101966
34
 81700916 1.618033989 2.618033989 4.236067977 6.854101966
35
 132194859 1.618033989 2.618033989 4.236067977 6.854101966
 
Golden Ratio
Square
of Golden Ratio
Cubic
of Golden Ratio

4th power
of Golden Ratio

 
When
f(0)=-7 and f(1)=-3		  
n
Sequence
f(n)/f(n-1)
f(n)/f(n-2)
f(n)/f(n-3)
f(n)/f(n-4)
1
 -7      
2
 -3 0.428571429    
3
 -10 3.333333333 1.428571429    
4
 -13 1.3 4.333333333 1.857142857  
5
 -23 1.769230769 2.3 7.666666667 3.285714286
6
 -36 1.565217391 2.769230769 3.6 12
7
 -59 1.638888889 2.565217391 4.538461538 5.9
8
 -95 1.610169492 2.638888889 4.130434783 7.307692308
9
 -154 1.621052632 2.610169492 4.277777778 6.695652174
10
 -249 1.616883117 2.621052632 4.220338983 6.916666667
11
 -403 1.618473896 2.616883117 4.242105263 6.830508475
12
 -652 1.617866005 2.618473896 4.233766234 6.863157895
13
 -1055 1.61809816 2.617866005 4.236947791 6.850649351
14
 -1707 1.618009479 2.61809816 4.23573201 6.855421687
15
 -2762 1.618043351 2.618009479 4.236196319 6.853598015
16
 -4469 1.618030413 2.618043351 4.236018957 6.854294479
17
 -7231 1.618035355 2.618030413 4.236086702 6.854028436
18
 -11700 1.618033467 2.618035355 4.236060825 6.854130053
19
 -18931 1.618034188 2.618033467 4.236070709 6.854091238
20
 -30631 1.618033913 2.618034188 4.236066934 6.854106064
21
 -49562 1.618034018 2.618033913 4.236068376 6.854100401
22
 -80193 1.618033978 2.618034018 4.236067825 6.854102564
23
 -129755 1.618033993 2.618033978 4.236068036 6.854101738
24
 -209948 1.618033987 2.618033993 4.236067955 6.854102053
25
 -339703 1.618033989 2.618033987 4.236067986 6.854101933
26
 -549651 1.618033989 2.618033989 4.236067974 6.854101979
27
 -889354 1.618033989 2.618033989 4.236067979 6.854101961
28
 -1439005 1.618033989 2.618033989 4.236067977 6.854101968
29
 -2328359 1.618033989 2.618033989 4.236067978 6.854101966
30
 -3767364 1.618033989 2.618033989 4.236067977 6.854101967
31
 -6095723 1.618033989 2.618033989 4.236067978 6.854101966
32
 -9863087 1.618033989 2.618033989 4.236067977 6.854101966
33
 -15958810 1.618033989 2.618033989 4.236067978 6.854101966
34
 -25821897 1.618033989 2.618033989 4.236067977 6.854101966
35
 -41780707 1.618033989 2.618033989 4.236067978 6.854101966
 
Golden Ratio
Square
of Golden Ratio
Cubic
of Golden Ratio

4th power
of Golden Ratio

 

Conclusion

  
  • As we see, no matter what numbers we put into f(0) and f(1), the ratios are approaching the same limits of the sequence as Fibonnaci Sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively.

>>> Conjecture : The reason why the ratios are approaching the same limits of the sequence as Fibonnaci Sequence, which are Golden Ratio, Square of Golden Ratio, Cubic Golden Ratio, 4th power of Golden Ratio, etc. respectively, is the ratios come from the recursion, not from the initial numbers.


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