Select which part of the sequence A(n) you want with the From and To boxes giving the starting and ending values of n. The series numbers are shown in the Sequence Values box and in the Sequence Plot they are translated into squares (give the size in pixels) on a "graph" where the direction of the graph's base line is selected (>=across, v=down, ^=up) and the numbers becomes distances from that base line.

The Spectrum and Signature numbers can be expressions such as **sqrt(2)** or **sin(Pi/3)**
(trig functions use radian meaure). **Phi** means (sqrt(5)+1)/2 and **phi** is (sqrt(5)-1)/2.

Click on a Sequence name button to see the Sequence Plot and Values.

Notes on the sequences themselves and what they mean are lower down on this page.

Sequence Plot:

- Vertical Para-Fib
- A(n) is the row number of n in the Wythoff array
- Horizontal Para-Fib
- A(n) is the column number of n in the Wythoff array
- R(n)
- is the number of sets of Fibonacci numbers (1,2,3,5,8,13...) that sum to n, e.g. 8 is 8 itself or 3+5 or 1+2+5. These are the Fibonacci representations of 8 and there are 3 of them, so the 8-th number in this series, R(8), is 3. No value is repeated repeated in the sums.
- Rabbit Sequence
- or the Golden String is a sequence of 0s and 1s with many relationships with the Fibonacci numbers and the golden ratios Phi and phi
- G(a,b,n)
- is the Generalized Fibonacci Sequence that starts from a and b
and subsequent values are the sum of the last two. G(0,1) gives 0,1,1,2,3,5,... which is
*the*Fibonacci sequence; G(2,1) is 2,1,3,4,7,11,... the Lucas Numbers and so on.- Mod m
- All of these are increasing sequences of number and interesting visualisation patterns are producing
by taking the
*remainders when divided by m*which is called**mod m**. To see the actual numbers in a G series, use a large molulus such as 1000. - #reps
- In the same way that R(n) counts the number of sets of Fibonacci numbers each summing to n,
this is the number of different sets of values in the G series that have a sum of 1, 2, 3, ,.. .
**#reps**means the*number of representations*.

- Spectrum of x
- .. is the series of the whole number parts of the multiples of x, i.e. { Int(n x) | n=1,2,3,...}.
The whole number part is just that part before the decimal point so that the whole number part of 3.872 is 3.
e.g. Phi = 1.6180339... so its multiples including itself are
Phi = 1.61803, 2 Phi = 3.23607, 3 Phi = 4.8541, 4 Phi = 6.47214, 5 Phi = 8.09017...

and so its spectrum is1, 3, 4, 6, 8, ...

Each real number has a unique spectrum. If the number x is not expressible as a fraction (x is irrational), the spectrum is called the Beatty Sequence of x.

More at mathworld - The signature of x
- Take all the multiples of the given number, x, and add on to each the whole numbers, c, in turn from c=1 and
we get a table of
values of c+dx. E.g. if x is Phi=1.618.. we have the table shown here:

c+dPhi d 1 2 3 4 c 1 2.62 4.24 5.85 7.47 2 3.62 5.24 6.85 8.47 3 4.62 6.24 7.85 9.47 4 5.62 7.24 8.85 10.47 5 6.62 8.24 9.85 11.47

We now sort the entries in the table into increasing order. The row values (c) of each entry in this ordered list is called the*signature of x*.

Sorting, with (c,d) being the row and column numbers, we have 2.62 at (*1*,1), 3.62 at (*2*,1), 4.24 at (*1*,2), 4.62 at (*3*,1), 5.24 at (*2*,2) and so on. The sequence of row numbers (c) in this list is the*signature of Phi*: 1,2,1,3,2,...

More at Mathworld .

© Dr Ron Knott for Casey Mongoven: version 7 February 2005