Assignment 12 :: Spreadsheets

Fibonacci Sequence

By Jamie K. York

Have you ever seen a pattern in nature? Perhaps you have seen one in a seashell or a cactus. Many of these patterns can be described mathematically using sequences.

The Fibonacci sequence is a series of numbers such that f(0) = 1, f(1) = 1, and f(n) = f(n-1) + f(n-2).  See the full list of numbers in the series below, up to f(30):

 n f(n) 0 1 1 1 2 2 3 3 4 5 5 8 6 13 7 21 8 34 9 55 10 89 11 144 12 233 13 377 14 610 15 987 16 1597 17 2584 18 4181 19 6765 20 10946 21 17711 22 28657 23 46368 24 75025 25 121393 26 196418 27 317811 28 514229 29 832040 30 1346269

In the sequence, there are multiple patterns that exist. LetŐs explore these together.

In a comparison of adjacent terms in the sequence, we can see that the ratio approaches a specific value as n increases.

 n f(n) Ratio of Adjacent Terms 0 1 1 1 1 0.50000000 2 2 0.66666667 3 3 0.60000000 4 5 0.62500000 5 8 0.61538462 6 13 0.61904762 7 21 0.61764706 8 34 0.61818182 9 55 0.61797753 10 89 0.61805556 11 144 0.61802575 12 233 0.61803714 13 377 0.61803279 14 610 0.61803445 15 987 0.61803381 16 1597 0.61803406 17 2584 0.61803396 18 4181 0.61803400 19 6765 0.61803399 20 10946 0.61803399 21 17711 0.61803399 22 28657 0.61803399 23 46368 0.61803399 24 75025 0.61803399 25 121393 0.61803399 26 196418 0.61803399 27 317811 0.61803399 28 514229 0.61803399 29 832040 0.61803399 30 1346269

Every Second Term

Next we can look at the ratio of every second term. Similarly to the ratio of adjacent terms, we see that this ratio approaches a new value as n increases.

 n f(n) Ratio of Every Second Term 0 1 0.5 1 1 0.333333333 2 2 0.4 3 3 0.375 4 5 0.384615385 5 8 0.380952381 6 13 0.382352941 7 21 0.381818182 8 34 0.382022472 9 55 0.381944444 10 89 0.381974249 11 144 0.381962865 12 233 0.381967213 13 377 0.381965552 14 610 0.381966187 15 987 0.381965944 16 1597 0.381966037 17 2584 0.381966001 18 4181 0.381966015 19 6765 0.38196601 20 10946 0.381966012 21 17711 0.381966011 22 28657 0.381966011 23 46368 0.381966011 24 75025 0.381966011 25 121393 0.381966011 26 196418 0.381966011 27 317811 0.381966011 28 514229 0.381966011 29 832040 30 1346269

Sequence Modifications

The Fibonacci sequence above is based on the fact that f(0) = 1 and f(1) = 1. What would happen to the pattern if these constants were altered? LetŐs consider other integers greater than 1.

If f(0) = 1 and f(1) = 3, this reflects a Lucas Sequence. However, with further analysis of the same two ratios, you will find that they approach the same values. The ratio of adjacent terms approaches 0.61803399 as n increases, and the ratio of every second term approaches 0.381966011.

 n f(n) Ratio of Adjacent Terms Ratio of Every Second Term 0 1 0.333333333 0.25 1 3 0.75000000 0.428571429 2 4 0.57142857 0.363636364 3 7 0.63636364 0.388888889 4 11 0.61111111 0.379310345 5 18 0.62068966 0.382978723 6 29 0.61702128 0.381578947 7 47 0.61842105 0.382113821 8 76 0.61788618 0.381909548 9 123 0.61809045 0.381987578 10 199 0.61801242 0.381957774 11 322 0.61804223 0.381969158 12 521 0.61803084 0.381964809 13 843 0.61803519 0.38196647 14 1364 0.61803353 0.381965836 15 2207 0.61803416 0.381966078 16 3571 0.61803392 0.381965986 17 5778 0.61803401 0.381966021 18 9349 0.61803398 0.381966008 19 15127 0.61803399 0.381966013 20 24476 0.61803399 0.381966011 21 39603 0.61803399 0.381966011 22 64079 0.61803399 0.381966011 23 103682 0.61803399 0.381966011 24 167761 0.61803399 0.381966011 25 271443 0.61803399 0.381966011 26 439204 0.61803399 0.381966011 27 710647 0.61803399 0.381966011 28 1149851 0.61803399 0.381966011 29 1860498 0.61803399 30 3010349

In further investigation, we can explore other modifications to the value of f(1) to find that these two ratios still remain unchanged.

f(1) = 4

 n f(n) Ratio of Adjacent Terms Ratio of Every Second Term 0 1 0.25 0.2 1 4 0.80000000 0.444444444 2 5 0.55555556 0.357142857 3 9 0.64285714 0.391304348 4 14 0.60869565 0.378378378 5 23 0.62162162 0.383333333 6 37 0.61666667 0.381443299 7 60 0.61855670 0.382165605 8 97 0.61783439 0.381889764 9 157 0.61811024 0.381995134 10 254 0.61800487 0.381954887 11 411 0.61804511 0.38197026 12 665 0.61802974 0.381964388 13 1076 0.61803561 0.381966631 14 1741 0.61803337 0.381965774 15 2817 0.61803423 0.381966102 16 4558 0.61803390 0.381965977 17 7375 0.61803402 0.381966024 18 11933 0.61803398 0.381966006 19 19308 0.61803399 0.381966013 20 31241 0.61803399 0.381966011 21 50549 0.61803399 0.381966012 22 81790 0.61803399 0.381966011 23 132339 0.61803399 0.381966011 24 214129 0.61803399 0.381966011 25 346468 0.61803399 0.381966011 26 560597 0.61803399 0.381966011 27 907065 0.61803399 0.381966011 28 1467662 0.61803399 0.381966011 29 2374727 0.61803399 30 3842389

f(1) = 10

 n f(n) Ratio of Adjacent Terms Ratio of Every Second Term 0 1 0.1 0.090909091 1 10 0.90909091 0.476190476 2 11 0.52380952 0.34375 3 21 0.65625000 0.396226415 4 32 0.60377358 0.376470588 5 53 0.62352941 0.384057971 6 85 0.61594203 0.381165919 7 138 0.61883408 0.382271468 8 223 0.61772853 0.381849315 9 361 0.61815068 0.382010582 10 584 0.61798942 0.381948986 11 945 0.61805101 0.381972514 12 1529 0.61802749 0.381963527 13 2474 0.61803647 0.38196696 14 4003 0.61803304 0.381965649 15 6477 0.61803435 0.38196615 16 10480 0.61803385 0.381965958 17 16957 0.61803404 0.381966031 18 27437 0.61803397 0.381966004 19 44394 0.61803400 0.381966014 20 71831 0.61803399 0.38196601 21 116225 0.61803399 0.381966012 22 188056 0.61803399 0.381966011 23 304281 0.61803399 0.381966011 24 492337 0.61803399 0.381966011 25 796618 0.61803399 0.381966011 26 1288955 0.61803399 0.381966011 27 2085573 0.61803399 0.381966011 28 3374528 0.61803399 0.381966011 29 5460101 0.61803399 30 8834629

We can also explore modifications to the value of f(0) to find that these two ratios still remain unchanged.

f(0) = 5

 n f(n) Ratio of Adjacent Terms Ratio of Every Second Term 0 5 5 0.833333333 1 1 0.16666667 0.142857143 2 6 0.85714286 0.461538462 3 7 0.53846154 0.35 4 13 0.65000000 0.393939394 5 20 0.60606061 0.377358491 6 33 0.62264151 0.38372093 7 53 0.61627907 0.381294964 8 86 0.61870504 0.382222222 9 139 0.61777778 0.381868132 10 225 0.61813187 0.382003396 11 364 0.61799660 0.381951731 12 589 0.61804827 0.381971466 13 953 0.61802853 0.381963928 14 1542 0.61803607 0.381966807 15 2495 0.61803319 0.381965707 16 4037 0.61803429 0.381966127 17 6532 0.61803387 0.381965967 18 10569 0.61803403 0.381966028 19 17101 0.61803397 0.381966005 20 27670 0.61803400 0.381966014 21 44771 0.61803399 0.38196601 22 72441 0.61803399 0.381966012 23 117212 0.61803399 0.381966011 24 189653 0.61803399 0.381966011 25 306865 0.61803399 0.381966011 26 496518 0.61803399 0.381966011 27 803383 0.61803399 0.381966011 28 1299901 0.61803399 0.381966011 29 2103284 0.61803399 30 3403185

f(0) = 10

 n f(n) Ratio of Adjacent Terms Ratio of Every Second Term 0 10 10 0.909090909 1 1 0.09090909 0.083333333 2 11 0.91666667 0.47826087 3 12 0.52173913 0.342857143 4 23 0.65714286 0.396551724 5 35 0.60344828 0.376344086 6 58 0.62365591 0.38410596 7 93 0.61589404 0.381147541 8 151 0.61885246 0.382278481 9 244 0.61772152 0.381846635 10 395 0.61815336 0.382011605 11 639 0.61798839 0.381948595 12 1034 0.61805140 0.381972663 13 1673 0.61802734 0.38196347 14 2707 0.61803653 0.381966982 15 4380 0.61803302 0.381965641 16 7087 0.61803436 0.381966153 17 11467 0.61803385 0.381965957 18 18554 0.61803404 0.381966032 19 30021 0.61803397 0.381966003 20 48575 0.61803400 0.381966014 21 78596 0.61803399 0.38196601 22 127171 0.61803399 0.381966012 23 205767 0.61803399 0.381966011 24 332938 0.61803399 0.381966011 25 538705 0.61803399 0.381966011 26 871643 0.61803399 0.381966011 27 1410348 0.61803399 0.381966011 28 2281991 0.61803399 0.381966011 29 3692339 0.61803399 30 5974330

Now, we can explore the modification of both variables to see that the ratios still remain constant.

f(0) = 5, f(1) = 5

 n f(n) Ratio of Adjacent Terms Ratio of Every Second Term 0 5 1 0.5 1 5 0.50000000 0.333333333 2 10 0.66666667 0.4 3 15 0.60000000 0.375 4 25 0.62500000 0.384615385 5 40 0.61538462 0.380952381 6 65 0.61904762 0.382352941 7 105 0.61764706 0.381818182 8 170 0.61818182 0.382022472 9 275 0.61797753 0.381944444 10 445 0.61805556 0.381974249 11 720 0.61802575 0.381962865 12 1165 0.61803714 0.381967213 13 1885 0.61803279 0.381965552 14 3050 0.61803445 0.381966187 15 4935 0.61803381 0.381965944 16 7985 0.61803406 0.381966037 17 12920 0.61803396 0.381966001 18 20905 0.61803400 0.381966015 19 33825 0.61803399 0.38196601 20 54730 0.61803399 0.381966012 21 88555 0.61803399 0.381966011 22 143285 0.61803399 0.381966011 23 231840 0.61803399 0.381966011 24 375125 0.61803399 0.381966011 25 606965 0.61803399 0.381966011 26 982090 0.61803399 0.381966011 27 1589055 0.61803399 0.381966011 28 2571145 0.61803399 0.381966011 29 4160200 0.61803399 30 6731345

Further Exploration

How would the pattern change if f(0) and f(1) were changed to non-integer values?

How would the pattern change if f(0) and f(1) were changed to irrational values?