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In the Fibonacci Sequence (1, 1, 2, 3, 5, 8, 13, ...), each term is the sum of the two previous terms (for instance, 2+3 = 5, 3+5 = 8, ...). As you go farther and farther in this sequence, the ratio of a term to the one before it gets closer and closer to the Golden Ratio.

To find the Fibonacci numbers in Pascal's triangle you have to go up at an angle along the "shallow diagonals": you're looking for 1, 1, 1+1, 1+2, 1+3+1, 1+4+3, 1+5+6+1.


Chebyshev polynomials (1,9,28,35,15,1) are also found in the shallow diagonals of Pascal's triangle. The polynomial with these coefficients is


References

  1. Ask Dr. Math Archives: Fibonacci Sequence/Golden Ratio
  2. Ask Dr. Math FAQ: Golden Ratio, Fibonacci Sequence
  3. MathPages, Kevin Brown: Polynomials From Pascal's Triangle


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