Fibonacci Numbers

F(n) = F(n-1) + F(n-2)
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

The Fibonacci sequence starts with 1, 1, and each subsequent number is the sum of the two before it. Named after Leonardo of Pisa (Fibonacci) who described it in 1202, the sequence had been known in Indian mathematics centuries earlier. Its ratios converge to the golden ratio phi, and it appears throughout nature wherever efficient packing occurs.

Fibonacci spiral: squares and quarter-circle arcs (like the nautilus)
21 13 8 5 3 2 1, 1, 2, 3, 5, 8, 13, 21 - each number = sum of the two before it
Fibonacci in Pascal's triangle: shallow diagonals sum to Fibonacci numbers
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 1 1+1=2 1+2=3 Each shallow diagonal sums to a Fibonacci number: 1, 1, 2, 3, 5, 8, 13...
Binet's formula: closed-form Fibonacci from the golden ratio
F(n) = (phi^n - psi^n) / sqrt(5) phi = (1+sqrt(5))/2 = 1.61803... psi = (1-sqrt(5))/2 = -0.61803... Because |psi| < 1, psi^n shrinks. F(n) is the nearest integer to phi^n / sqrt(5).
Golden Ratio phi → Golden Angle → Irrational Numbers →
Related topics
Phi Golden Angle Tribonacci
Key facts about Fibonacci Numbers

The Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34... is defined by F(n) = F(n-1) + F(n-2). Named after Leonardo of Pisa who introduced it to Europe in 1202, the sequence was known in Indian mathematics from at least the 6th century. Consecutive Fibonacci ratios converge to the golden ratio phi. The sequence appears in sunflower seed spirals, pinecone bracts, pineapple scales, and the branching of trees. Binet's formula gives an exact closed form: F(n) = (phi^n - psi^n) / sqrt(5).

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