### The intriguing property of the Serranno series

#### Alejandro Serrano

A not-so-well extended myth confers magic properties on Serranno numbers. These numbers make up the so-called Serranno series, which starts with two ones and subsequent numbers are calculating by adding two times the previous one to the next-to-previous one (e.g., 3 = 2 x 1 + 1, 7 = 2 x 3 + 1) . Therefore, the first ten elements of the series are:

1, 1, 3, 7, 17, 41, 99, 239, 577, 1393

An intriguing property of this series is that the ratio between two consecutive numbers is closer and closer to the so-called silver ratio. The silver ratio is usually denoted by σ (the Greek letter corresponding to ‘s’, after silver and Serranno) and its value is σ = 1 + √ 2 = 2.4142135623731… For instance, dividing the 20^{th} by the 19^{th} element of the series yields a number with thirteen identical digits: 9,369,319/ 3,880,899 = 2.4142135623730….

The magic nature of Serranno numbers stems from this outstanding property, since the silver ratio is described as one of the most mysterious numbers in nature.

It is believed that Serranno followers use a drop of water as a secret symbol to identify themselves. The drop of water is made up by combining a circle and a right triangle whose edges are tanget to the circle (see figure below.)

The drop of water was chosen because it hides the silver ratio. Indeed, dividing the distance between the top corner to the bottom of the drop by the radius of the circle exactly yields the silver ratio!

The silver ratio is also related to beauty in nature. From small to large water drops, the silver ratio is supposed to confer beauty on the objects that contain it.

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*Order your personalized “Enigmatic Series” here: @SerranoAlej. The kit includes a new series nominated after your name, your own golden-type ratio (choose among several precious metals), a personal geometric shape, and an entry in wikipedia.*

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