Brunardot Series     

The Brunardot Series is an additive, unending series, of unending sequences, that have a Natural integer for the second term; and, the third term is the square of the first term and, as with successive terms, is the sum of the preceding two terms.

Brunardot Series' sequences:

Second Seq: 1, 0, 1, 1, 2, 3, 5, 8...

 

Third Seq: Phi, 1, Phi +1, Phi + 2, 2 Phi + 3...
Fourth Seq: 2, 2, 4, 6, 10, 16, 26, 42...
Fifth Seq: (SqRt(13) + 1)/2, 3...
Sixth Seq: (SqRt(17) + 1)/2, 4...
Sixth Series: (SqRt(21) + 1)/2, 5...
Seventh Series: 3, 6, 9, 15, 24, 39, 63, 103...
Eighth Series: (SqRt(29) + 1)/2, 7...

 . . .

Thirteenth Series:
4, 12, 16, 28, 44, 72, 116, 188...

 . . .

Twenty-first Series: 5, 20, 25, 45, 70, 115, 185, 300...


 *

 

The Second Series can also be written:

(SqRt(5) + 1)/2, 1...

The first Brunardot Series, beginning with the third term, is the Fibonacci series (1,1,2,3,5...).  Because the Fibonacci series is commonly found in nature; and, the Brunardot Series is a Natural Series, arguably, it can be said that the two initial terms of the first Brunardot Series (1 and 0) properly belong to the Fibonacci Sequence.  The Fibonacci Sequence, so completed: 1, 0, 1, 1, 2, 3, 5..., is referred to as the Revised Fibonacci Sequence.

The first two terms of a Brunardot Series are the Pulser of Brunardot Ellipses.

The first term of the Second Brunardot Series begins with Phi,
F, the Golden Ratio, which is: (SqRt(5) + 1)/2.  The Second Brunardot Series is referred to as the Golden Series.

Thus, Brunardot Ellipses, defined by the Brunardot Series, closely relate the Fibonacci Series and Phi, The Golden Ratio.

The formulas for the first and third terms of the Brunardot Series in terms of the second term, s, are as follows:

First Term: (SqRt(4s + 1) + 1)/2
Third Term: (SqRt(4s + 1) + 1)/2 + s


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