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Fibonacci Sequence: First 100 Numbers Matrix

Fibonacci Sequence: First 100 Numbers Matrix

Fibonacci Sequence: First 100 Numbers Matrix

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The Fibonacci sequence starts 1, 1, 2, 3, 5, 8, where each number is the sum of the two before it. This matrix lists the exact first 100 values with the golden ratio they approach, plus a live F(n) calculator.

RecurrenceF(n) = F(n−1) + F(n−2)Each number is the sum of the previous two
Start ValuesF(0) = 0, F(1) = 1This table lists F(1) to F(100)
Golden Ratioφ = (1 + √5) / 2About 1.6180339887
Binet's FormulaF(n) = (φn − ψn) / √5Exact on paper, inexact in 64-bit floats
F(50)12,586,269,02511 digits, fits in 64-bit
F(100)354,224,848,179,261,915,07521 digits, needs BigInt

Fibonacci Calculator

Tip: open this page with ?n=50 to start with F(50) and highlight its row (the table lists F(1) to F(100); the calculator is exact up to F(1000) using BigInt).

First 100 Fibonacci Numbers: Exact Values and Golden Ratio
Index (Fn) Exact Fibonacci Value Golden Ratio (φ) Ratio: Fn / Fn−1 Copy
F11n/a (F0 = 0)
F211.0000000000
F322.0000000000
F431.5000000000
F551.6666666667
F681.6000000000
F7131.6250000000
F8211.6153846154
F9341.6190476190
F10551.6176470588
F11891.6181818182
F121441.6179775281
F132331.6180555556
F143771.6180257511
F156101.6180371353
F169871.6180327869
F1715971.6180344478
F1825841.6180338134
F1941811.6180340557
F2067651.6180339632
F21109461.6180339985
F22177111.6180339850
F23286571.6180339902
F24463681.6180339882
F25750251.6180339890
F261213931.6180339887
F271964181.6180339888
F283178111.6180339887
F295142291.6180339888
F308320401.6180339887
F3113462691.6180339888
F3221783091.6180339887
F3335245781.6180339887
F3457028871.6180339887
F3592274651.6180339887
F36149303521.6180339887
F37241578171.6180339887
F38390881691.6180339887
F39632459861.6180339887
F401023341551.6180339887
F411655801411.6180339887
F422679142961.6180339887
F434334944371.6180339887
F447014087331.6180339887
F4511349031701.6180339887
F4618363119031.6180339887
F4729712150731.6180339887
F4848075269761.6180339887
F4977787420491.6180339887
F50125862690251.6180339887
F51203650110741.6180339887
F52329512800991.6180339887
F53533162911731.6180339887
F54862675712721.6180339887
F551395838624451.6180339887
F562258514337171.6180339887
F573654352961621.6180339887
F585912867298791.6180339887
F599567220260411.6180339887
F6015480087559201.6180339887
F6125047307819611.6180339887
F6240527395378811.6180339887
F6365574703198421.6180339887
F64106102098577231.6180339887
F65171676801775651.6180339887
F66277778900352881.6180339887
F67449455702128531.6180339887
F68727234602481411.6180339887
F691176690304609941.6180339887
F701903924907091351.6180339887
F713080615211701291.6180339887
F724984540118792641.6180339887
F738065155330493931.6180339887
F7413049695449286571.6180339887
F7521114850779780501.6180339887
F7634164546229067071.6180339887
F7755279397008847571.6180339887
F7889443943237914641.6180339887
F79144723340246762211.6180339887
F80234167283484676851.6180339887
F81378890623731439061.6180339887
F82613057907216115911.6180339887
F83991948530947554971.6180339887
F841605006438163670881.6180339887
F852596954969111225851.6180339887
F864201961407274896731.6180339887
F876798916376386122581.6180339887
F8811000877783661019311.6180339887
F8917799794160047141891.6180339887
F9028800671943708161201.6180339887
F9146600466103755303091.6180339887
F9275401138047463464291.6180339887
F93122001604151218767381.6180339887
F94197402742198682231671.6180339887
F95319404346349900999051.6180339887
F96516807088548583230721.6180339887
F97836211434898484229771.6180339887
F981353018523447067460491.6180339887
F992189229958345551690261.6180339887
F1003542248481792619150751.6180339887
Fibonacci Integer Overflow Limits by Data Type
Data Type Maximum Value Largest n That Fits F(n) at That Index First Overflow Copy
32-bit signed int2147483647F(46)1836311903F(47)
32-bit unsigned int4294967295F(47)2971215073F(48)
64-bit signed int (long)9223372036854775807F(92)7540113804746346429F(93)
64-bit unsigned int18446744073709551615F(93)12200160415121876738F(94)
JavaScript Number (safe integer)9007199254740991F(78)8944394323791464F(79)
Fibonacci Algorithms: Complexity and Copy-Ready Code
Method Language Time Space Best For Copy Code
Naive recursionPythonO(2n)O(n)Teaching only; F(50) needs over 40 billion calls
MemoizationPythonO(n)O(n)Simple recursive solution in interviews (n up to about 900)
Iterative loopPythonO(n)O(1)Default answer: fast, tiny, no recursion limit
Fast doublingPythonO(log n)O(log n)Huge n such as F(1,000,000)
Iterative BigIntJavaScriptO(n)O(1)Exact values beyond F(78) in the browser or Node

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Frequently Asked Questions

Question Answer
What are the first 10 Fibonacci numbers?Starting from F(0) = 0 the first 10 are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. This page lists F(1) to F(100), where F(1) = 1 and F(2) = 1.
What is F(100), the 100th Fibonacci number?F(100) = 354224848179261915075, a 21-digit number. It is too large for a 64-bit integer, so use BigInt, Python int or another arbitrary-precision type.
How do you compute Fibonacci numbers efficiently?Use an iterative loop (O(n) time, O(1) space) or fast doubling (O(log n)). Naive recursion is O(2n): F(50) alone needs 40,730,022,147 function calls.
Why does F(n)/F(n-1) approach the golden ratio?The ratio of consecutive Fibonacci numbers converges to φ = (1 + √5) / 2 = 1.6180339887... It already matches φ to 10 decimals at F(26)/F(25).
Which Fibonacci number overflows a 64-bit integer?F(93) = 12200160415121876738 overflows a signed 64-bit integer (the largest that fits is F(92) = 7540113804746346429). F(94) overflows an unsigned 64-bit integer.
Is there a closed formula for the nth Fibonacci number?Yes, Binet's formula: F(n) = (&φ;n − &ψ;n) / √5, with ψ = (1 − √5) / 2. With 64-bit floating point it rounds to the wrong integer from n = 71, so use integer methods for exact results.

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