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Any four consecutive Fibonacci numbers F n , F n +1 , F n +2 and F n +3 can also be used to generate a Pythagorean triple in a different way: [55]

Example 1: let the Fibonacci numbers be 1, 2, 3 and 5. Then:

Since F n is asymptotic to ${\phi }^{n}/\sqrt{5}$ , the number of digits in F n is asymptotic to $n{\mathrm{log}}_{10}\phi \approx 0.2090n$ . As a consequence, for every integer d > 1 there are either 4 or 5 Fibonacci numbers with d decimal digits.

More generally, in the base b representation, the number of digits in F n is asymptotic to $n{\mathrm{log}}_{b}\phi$ .

The Fibonacci numbers are important in the computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers. [56]

Brasch et al. 2012 show how a generalised Fibonacci sequence also can be connected to the field of economics. [57] In particular, it is shown how a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.

### Stories of learning and leading

by George

Within leadership, one of my focuses is to take the complex and try to make it simple. Not “simple” in the delivery and the work, but in the messaging to create a clearer vision of what is essential to focus on now and in the future. In my mind, I keep coming back to three main ideas.

Simple concepts but massive undertakings in our work. But what does each mean when you dig in?

The Environment is about the space(s) around us and the “stuff” that is needed to serve learners.

The Experience is about what we design for learning and what we empower others to create for themselves.

The Energy is what we bring in ourselves to our schools and what others help unleash in others.

As I write about each one, I understand there is a definite overlap. It is hard to create a fantastic for our students and our classrooms when our is either lacking or pulling us in a negative direction. The can be adversely affected by a lackluster experience. Many schools who have access to resources, that create an experience that is less than ideal.

Can a great learning experience in school happen where one is lacking? To some extent, yes, but I would say that if all three are a focus, the chances of success become greater. I did, however, list the ideas in what I believe to be in the opposite direction of where I see them being most valuable. Many educators are exerting tremendous energy to overcome a bad experience in their learning, as well as less than ideal conditions in the environment. But if the first two “E’s” are there, it can enhance the an educator brings to the profession. Education is a career pathway that can be so emotionally, mentally, and sometimes even physically exhausting, so it has been incredible to see how more schools are focusing on supporting teachers to take care of themselves, so they can excel at what they are doing.

What is important is that each one of these “E’s” is built within and with your school or organizational culture. What are the elements within each that can most support the people serve? How do you enhance each focus area while still working within certain constraints that are often out of the control of schools? Where do you excel and where do you lack, and why?

I will be looking at these three “E’s” more in the future but would love your thoughts. Again, this is not to say being an educator is simple, but more to streamline a focus on big ideas and how important they are to supporting our learners and helping them excel. I would love your thoughts not only each element listed but examples within each that support the ideas.

I am writing to learn, not necessarily to share my learning. I am trying to work out these ideas through my writing so any feedback would be appreciated.

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