Archive for the ‘Mathematics’ Category

Fractional Fibonacci Numbers

Thursday, 7 June 2018

Traditional Formula

There is a well-known formula for the Fibonacci numbers

\displaystyle F_n = \dfrac{\sqrt{5}}{5} \left( \varphi^n - (-\varphi)^{-n} \right)

where

\displaystyle \varphi = \dfrac{\sqrt{5} + 1}{2} \approx 1.618^{+}

is the golden ratio.

It turns out that there is a way to find F_x for when x is not an integer, but the values are complex rather than real.

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“New Approach to Sums of Powers” — Headlines and Examples

Thursday, 7 June 2018

As the article on sums of powers was rather long and dense, I thought that it would be worth giving a summary of the main results separately.

I will also show the formulae in action with a worked example.

Indirect, Simple Formulae

In the main article, I show that

\displaystyle {\bigoplus_{m=1}^{n}}{}^{(t)} \; m^k = \sum_{j=0}^{k} \left< \begin{array}{c} k \\ j \\ \end{array} \right> \triangle_{k+t}(n-j)

This formula is essentially a polynomial of rising factorial powers.

Special Cases

Perhaps the most important and useful formulae from the main article are

\displaystyle n^k = \sum_{j=0}^{k} \left< \begin{array}{c} k \\ j \\ \end{array} \right> \triangle_{k}(n-j)

and

\displaystyle \sum_{m=1}^{n} m^k = \sum_{j=0}^{k} \left< \begin{array}{c} k \\ j \\ \end{array} \right> \triangle_{k+1}(n-j)

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PDF version of A New Approach to the Sums of Powers

Thursday, 10 May 2018

A PDF version of the previous post is here.

A New Approach to the Sums of Powers

Thursday, 10 May 2018

In the conventional approach to summing powers, that is, finding a polynomial expression for \sum_{h=1}^{n} h^k, the coefficients that arise seem to have no pattern. It had always seemed to me that it ought not to be hard to find such expressions with an elementary approach.

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Product Formulae for the Fibonacci Numbers

Monday, 7 May 2018

There is a well-known formula for the Fibonacci numbers

\displaystyle F_n = \dfrac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}}

where

\displaystyle \varphi = \dfrac{1-\sqrt{5}}{2} \approx 1.618^{+}

However, I was surprised to find that there are also product formulae involving trigonometric functions.

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Tables for the Regular Polyhedra

Saturday, 23 July 2016

For quite some time now, I have been looking in books and online for a set of tables with formulae for conversion between various measures of the platonic solids (the regular polyhedra). None quite fitted my requirements, and so I created my own.

My requirements included:

  • The formulae should all be of a similar form.
  • Where there is a change of dimension, formulae should be given both in terms of the source and the target dimensons.
  • No formula should have a surd (root) in the denominator.
  • The terms in a surd should have reduced factors. (So, in particular, any integer under a square root should be square-free.)

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The Partition Sum of Powers Theorem

Tuesday, 8 March 2016

The set of numbers {S = \{ 0, 1, 2, \dots, 2^{n+1}-1 \}} can be partitioned into two subsets of the same size, such that the two sets have equal sums, sums of squares, sums of cubes, …, up to sums of {n}th powers.

For example, for {n=2}:

\displaystyle S = \{ 0, 1, 2, 3, 4, 5, 6, 7 \}

can be partitioned as

\displaystyle A = \{ 0, 3, 5, 6 \}, B = \{ 1, 2, 4, 7 \}

so that

\displaystyle |A| = |B| = 4

\displaystyle 0 + 3 + 5 + 6 = 1 + 2 + 4 + 7 = 14

and, lastly,

\displaystyle 0^2 + 3^2 + 5^2 + 6^2 = 1^2 + 2^2 + 4^2 + 7^2 = 70

Amazingly, this can be done for any non-negative integer {n}.

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QiX

Tuesday, 24 November 2015

This is a link to a historical project of mine, hosted on Albert Gräf’s project page.

QiX is a library for Albert Gräf’s Q programming language adding support for univariate polynomials.

There is full documentation available.

Q+Q

Tuesday, 24 November 2015

This is a link to a historical project of mine, hosted on Albert Gräf’s project page.

Q+Q is a library for Albert Gräf’s Q programming language adding support for the rational numbers, ℚ.

There is full documentation available.

Generating Approximate Pythagorean Angles (ADDENDUM) – Simplified Method

Tuesday, 20 October 2015

In a recent post I described a method of generating the simplest primitive Pythagorean triple (a,b,c) where one of the angles of the triangle with sides a, b and c is θ° to within some (small) error bound Δθ°.

One of the steps was, given the cosine C of the angle [from step (2)]:

(3) Calculate the Farey ratio approximant …

R = \sqrt{\dfrac{1-C}{1+C}}

Now, that function

f(C) = \sqrt{\dfrac{1-C}{1+C}}

seemed semi-familiar, resembling functions that occur in trigonometric or hyperbolic identities.
An example is:

\cos( \tan^{-1}(t)) = \sqrt{\dfrac{1}{t^2 + 1}}

A little further investigation, and reading around, including the Wikipedia articles on trigonometric identities, and in particular on those of the tangent half-angle, revealed that the Farey ratio approximant does in fact correspond directly to a simple trigonometric function of the angle:

\tan \left( \dfrac{\theta}{2} \right) = \dfrac{1 - \cos \theta}{\sin \theta} = \dfrac{1 - \cos \theta}{\sqrt{1 - \cos^2 \theta}} = \dfrac{\sqrt{(1 - \cos \theta)^2}}{\sqrt{(1 - \cos \theta)(1 + \cos \theta)}} = \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}}

The slightly simplified method follows.

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