This page contains links to pages for a variety of integer sequences that
have been converted to "shaft arithmetic" for
use in making threading
and treadling sequences (the use of "shaft" is arbitrary; the concept
applies equally well to treadles).
Repeats in sequences are noted by enclosing brackets.
For example, [1 2 3 2] is
a repeat, which when carried out gives the sequence
1 2 3 2 1 2 3 2 1 2 3 2 ... .
A program looks for repeats. If no repeat is detected, the first
100 values are given.
Note: Repeats are determined by a heuristic method. In some cases
they may be incorrect. In addition, very long repeats may not be
found from the number of terms computed. In cases where no repeat is
detected, there may or may not be a repeat. In fact, some sequences
given are known not to have repeats. The program has no knowledge of
these and simply reports that no repeat was detected.
The designation s= gives the number of shafts/treadles. Grid
plots for 100 shafts/treadles are given after each sequence.
The
On-Line Encyclopedia of Integer Sequences (EIS)
sequence number, if there is one, is given at the end of the section,
followed by links to relevant Web pages.
If you look at a sequence by its A number at the EIS site, you generally
will find references.
The "formerly" M numbers refer to sequences in the book The Encyclopedia of Integer Sequences.
More sequences will be added from time to time.
The sequence of positive integers (natural numbers)
is the most
basic of all integer sequences. In shaft arithmetic, it produces an
ascending (upward) straight draw.
EIS number: A000027
(formerly M0472)
The Fibonacci sequence is arguably the most
studied of all integer sequences. It has many important roles in
mathematics and aesthetics.
The first two terms of the Fibonacci sequence are 1 and 1. Each
subsequent term is the sum of the preceeding two. Written as a
recurrence, it is given by
a(n) = a(n - 1) + a(n - 2)
Note: The first two values sometimes are given as 0 and 1;
the resulting sequence is the same as for 1 and 1, except it starts
with 0.
There are many related sequences.
EIS number: A000045
(formerly M0692)
Links:
introduction
Lucas numbers
formulas
Fibonacci numbers and art
The primes are positive integers that are evenly
divisible only by themselves and 1. 1 is excluded for technical reasons;
the first and only even prime is 2. There are infinitely many primes.
They play important roles in number theory and cryptography and are
the subject of intense, on-going investigation. They also provide
fertile ground for recreational mathematics. Although there are
patterns in the prime sequence, these patterns are not regular and there is no
repeat.
EIS number: A000040
(formerly M0652)
The "multi" sequence consists of i
repetitions of each integer i: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... .
In shaft arithmetic, they produce interesting geometric weave structures.
Since the subsequences for each integer get longer and longer, they
do not produce repeats in shaft arithmetic.
EIS number: A000217
The Pell sequence starts with 1 and 2 and then is given by the recurrence
a(n) = 2 * a(n - 1) + a(n - 2)
Note: The first two values sometimes are given as 0 and 1;
the resulting sequence is the same as for 1 and 2, except it starts
with 0.
EIS number: A000129
(formerly M1413)
The Perrin sequence is starts with 0, 3, and 2, and then is given by the recurrence
a(n) = a(n - 2) + a(n - 3)
EIS number: A001608
(formerly M0429)
The "chaotic" sequence is starts with 1, 1, and 2, and then is given by the (nested) recurrence
a(n) = a(n - a(n - 1)) + a(n - a(n - 2))
There are many related sequences.
EIS number: A005185
(formerly M0438)
The triangular numbers are given by the formula
n * (n + 1) / 2
EIS number: A000217 (formerly M2535)
The pentagonal numbers are given by the formula
n * (3 * n - 1) / 2
EIS number: A000326
(formerly M3818)
The septagonal numbers are given by the formula
n * (5 * n - 1) / 2
EIS number: A000566
(formerly M1826)
The undecagonal (11-polygonal) numbers are given by the formula
n * (9 * n - 1) / 2
The versum (reverse and add)
sequence starting with 1.
versum sequences start with a number, reverse the order of the
digits, and add the result to the number. For example, the versum
sequence starting at 1 is 1, 2, 4, 8, 16, 77, 154, 605, ... .
The versum sequence starting with 196.
The Connell sequence is given by the formula
a(n) = 2 * n - [1 + sqrt(8 * n - 7)) / 2)]where [r] stands for the integer part of r, as in [4.3] = 4.
EIS number: A001614
(formerly M0962)
The Vishwanath sequences are like the Fibonacci sequence, except at every step, one of the following rules is chosen at random:
a(n) = a(n - 1) + a(n - 2)
a(n) = a(n - 1) - a(n - 2)
a(n) = -a(n - 1) + a(n - 2)
a(n) = -a(n - 1) - a(n - 2)
Since the rule to apply is picked at random, there is no one
Vishwanath sequence; in fact there are infintely many. Note also that
terms may be negative.
The lower Wythoff sequence (Beatty sequence of Type 1) is given by the formula
a(n) = [n * tau]where [r] stands for the integer part of r, as in [4.3] = 4, and tau is the golden ratio = 1.6180339887 ... .
EIS number: A000201
(formerly M2322)
The upper Wythoff sequence (Beatty sequence of Type 2) is given by the formula
a(n) = [n * tau ^ 2]where [r] stands for the integer part of r, as in [4.3] = 4, and tau is the golden ratio = 1.6180339887 ... .
EIS number: A001950
(formerly M1332)
The signature sequence of pi.
See
Eric Weisstein's definition of signature sequences.
EIS number:
A023133
The signature sequence of tau.
See
Eric Weisstein's definition of signature sequences.
The signature sequence of e.
See
Eric Weisstein's definition of signature sequences.
EIS number:
A023123
The signature sequence of sqrt(2).
See
Eric Weisstein's definition of signature sequences.
EIS number:
A007336
The signature sequence of e/pi.
See
Eric Weisstein's definition of signature sequences.
The Farey fraction numerator sequences.
The Farey fraction denominator sequences.
e-mail: ralph@cs.arizona.edu
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