What is a Fibonacci sequence

Fibonacci sequence

  • Fibonacci sequence (3:27 minutes)
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introduction

The Fibonacci sequence is an infinite sequence of numbers, where the following number is obtained by adding the two previous numbers:

$$ 0, \ enspace \, \, \, 1, \ enspace \, \, \, 1, \ enspace \, \, \, 2, \ enspace \, \, \, 3, \ enspace \, \, \, 5, \ enspace \, \, \, 8, \ enspace \, \, \, 13, \ enspace \, \, \, \ dots $$

Leonardo Fibonacci used this sequence in 1202 to describe the growth of a rabbit population.

Recursive formula

The Fibonacci sequence can be calculated using the following recursive formation law and the initial values ​​\ (f_0 \) and \ (f_1 \).

$$ f_0 = 0 \ qquad \ text {and} \ qquad f_1 = 1 $$

Each additional number is the sum of its two predecessors:

$$ f_n = f_ {n-1} + f_ {n-2} \ qquad \ text {for} \ qquad n \ geq 2 $$

table

The following table contains the Fibonacci numbers for \ (n \ leq 50 \):

\ (n \)\ (f_n \)
00
11
21
32
43
55
68
713
821
934
1055
1189
12144
13233
14377
15610
16987
171.597
182.584
194.181
206.765
2110.946
2217.711
2328.657
2446.368
2575.025
26121.393
27196.418
28317.811
29514.229
30832.040
311.346.269
322.178.309
333.524.578
345.702.887
359.227.465
3614.930.352
3724.157.817
3839.088.169
3963.245.986
40102.334.155
41165.580.141
42267.914.296
43433.494.437
44701.408.733
451.134.903.170
461.836.311.903
472.971.215.073
484.807.526.976
497.778.742.049
5012.586.269.025

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