## A Question of Numbers

**Brian Hayes**

Note: This document is available in other formats.

In my daydream, Neil Sloane and Simon Plouffe are contestants on
"Jeopardy," the TV game show. Sloane picks the category "Integer Sequences"
for $400, and Alex Trebek reads the answer: "1, 1, 2, 3, 5, 8, 13, 21...."
Sloane instantly supplies the question: "What are the Fibonacci numbers?"
Later it is Plouffe's turn, and he selects "Real Numbers" for $1,000.
Trebek reads out an answer: "1.618033989," and Plouffe responds with the
question: "What is phi, or the golden mean--the limiting value of the ratio
of successive Fibonacci numbers?"

In real life Sloane and Plouffe are not competitors but collaborators.
Sloane is a mathematician at AT&T Bell Laboratories, well known for his
work in graph theory, combinatorics and geometry. He is also the author of
the *Handbook of Integer Sequences*, a compendium of some 2,300 sequences, published in 1973. Plouffe, a mathematician now at Simon Fraser University in British Columbia, is another collector of numbers and sequences, who
volunteered a few years ago to help revise and expand the *Handbook*. Sloane
and Plouffe are coauthors of the new edition, published last year as the
*Encyclopedia of Integer Sequences*. It is a much-enlarged and enriched work, with more than 5,400 entries.

Sloane's sequence database is also accessible by electronic mail. If you
send a message to the Internet address sequences@research.att.com with the
text "lookup 1 1 2 5 14 42 132 429," you will receive a reply (typically a
few minutes later) identifying the sequence as the Catalan numbers, which
turn up in a surprising variety of combinatorial contexts
The information returned by the e-mail service typically includes
the initial terms of the sequence, a formula or generating function, a
terse description, and references to the literature. Sloane has also set up
a higher-power sequence server, which I'll discuss below.

Catalan numbers

Following his work on the *Encyclopedia of Integer Sequences*,
Plouffe has gone on to develop an analogous Internet server for real
numbers, called the
Inverse Symbolic Calculator,
or ISC. The calculator is
"inverse" in the sense that you give it a number and ask where the number
might have come from, rather than giving it a formula and requesting a
solution. You do not ask the calculator for the value of *e*/pi + 1;
you supply the numerical result 1.8652559794322, and the program suggests
this expression as one possible source.

**The Sequence of Sequences**

Network access to these resources adds a great deal of utility and
convenience, but I must pause to mention some special charms of the
old-fashioned, paper-and-ink version of the *Encyclopedia of Integer
Sequences*. There is no mathematical table I have read with greater
pleasure. Where the on-line service facilitates reference, the printed book
encourages browsing. As you leaf through the pages, you come upon sequences
you would not have thought to look up. For example, there is the sequence
designated M2675, which begins 1, 3, 7, 19, 53, 149, 419...; each element
*a*(*n*) is the number of stable towers that can be built from *n* Lego blocks. M1041, beginning 1, 2, 4, 7, 11, 16, 22..., is described as the maximum
number of pieces you can get from a pancake by slicing it with *n* cuts.
Sequence M0255, whose first few terms are 0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5,
4..., turns out to be the minimum number of multiplications needed to
compute an *n*th power.

Browsing also offers glimpses into some of the far-flung areas of
mathematics and science that are prolific generators of integer sequences.
Number theory and combinatorics are naturally well represented, but there
are also lots of examples from switching theory and circuit design
(combinations of Boolean functions), chemistry (numbers of alkanes, ethers,
esters, etc., with *n* carbon atoms) and a few from physics (Feynman diagrams with *n* vertices) and biology (secondary structures of RNA with *n* nucleotides). Here are a few of the more conspicuous landmarks among the sequences.

Sloane is particularly fond of self-generating and self-referential
sequences. In M0257, for instance, *a*(*n*) gives the number of
times that *n* appears in the sequence. The initial terms are 1, 2, 2,
3, 3, 4, 4, 4, 5, 5, 5.... Even cuter is Sloane's personal favorite, M4780,
which begins 1, 11, 21, 1211, 111221.... I will allow my readers to puzzle
this one out for themselves. (Here's a clue).

There are some outright jokes, such as M4961: 1, 15, 29, 12, 26, 12, 26,
9, 23, 7, 21.... The description given is "Dates at fortnightly intervals
from Jan. 1." Finite sequences are supposed to be excluded, but a few have
sneaked in. Sequence M3296 begins 1, 4, 7, 9, 11, 12, 14, 16... and is
given in its entirety in the *Encyclopedia*. Hint: It ends at 92 in
nature but goes on to at least 106 in the laboratory. Banned from the
printed volume but recently added to the on-line database are some other
well-known "dumb" sequences, such as lists of stops on various subway
lines.

Another benefit of browsing the printed page is that it sets one to
musing about the sequence of sequences. The arrangement of the
*Encyclopedia* at first seems fairly peculiar. The sequences are in a
lexicographic order that ignores any leading terms less than 2; thus 1, 2,
3, 4... precedes 0, 1, 2, 4, 8... but follows 2, 2, 4, 4, 6.... The reason
for this arrangement is that the starting point of many sequences is
uncertain or ambiguous. (Do the Fibonacci numbers begin 0, 1, 1, 2... or 1,
1, 2... or 1, 2...?) But solving one problem introduces other oddities.
What becomes of sequences that consist entirely of 0s and 1s? A few of
these have simply been placed ahead of the "official" table, whose first
entry is the sequence 2, 0, 0, 0.... Others have been transformed into
sequences of 1s and 2s.

No negative integers appear in the table. Hence there can be no infinite
monotonically decreasing sequence. I was mildly surprised that I had to
thumb through more than 200 sequences before coming to the first strictly
nondecreasing example. This is sequence M0208, and from the rule for
sorting the sequences you may be able to guess its identity: the all-2s
sequence, 2, 2, 2, 2.... In the printed *Encyclopedia* the very next
sequence, M0209--the slowest-growing increasing sequence--is the rather
interesting entry 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3,
3, 3, 3, 3, 3, 4, 4, 4, 5.... It counts the ways of partitioning a number
into cubes. When I mentioned this juxtaposition to Sloane, he pointed out
that his file of new sequences--those added to the collection since the
*Encyclopedia* was published--includes a few items that grow faster
than M0208 but more slowly than M0209. One example is the sequence in which
*a*(*n*) is the integer nearest to the base-10 logarithm of *n*; it
includes 285 2s followed by 2,846 3s.

**A Walk in Manhattan**

The primary use envisioned for both the printed *Encyclopedia* and the
Internet sequence server is reference. In the course of your work you
encounter a series of integers, and you want to know whether anyone has
seen them before and studied their properties. I have had occasion to
consult the server in just this way--although the sequences have come more
from play than from work.

When I was a commuter in New York, I walked a diagonal route across
midtown Manhattan morning and evening. I used to wonder how many different
paths I could find through the grid of east-west and north-south streets
without taking extra steps. Eventually I grew curious enough to calculate
some solutions. For a square lattice of *n*-by-*n* blocks, any minimum-length
diagonal path is necessarily 2*n* blocks long. How many such paths are there? For the 1-by-1 lattice the answer is 2: You can go east and then north or
you can go north and then east. In a two-block square there are six paths,
and in a 3-by-3 block there are 20. The sequence of values I calculated
begins like this: 2, 6, 20, 70, 252, 924, 3432, 12870, 48620....

Do those numbers look familiar? They should. They are prominent members
of one of the most ancient and famous families of numerical sequences. And
yet I did not identify them until several years later, when I had a chance
to submit them to Sloane's sequence server. The answer came back
immediately: They were recognized as sequence M1645, the central binomial
coefficients, the numbers that run down the middle of Pascal's triangle. My
reaction was "Of course! Why didn't I see it all along?" But I doubt that I
would have made the connection without help.

Manhattan paths

The discovery was a productive one, which led not just to an answer but
to an insight. I saw that counting the shortest diagonal paths for all
rectangular lattices--not just the square ones--would fill in the rest of
Pascal's triangle. Each row of the triangle consists of all the lattices
with a given minimum diagonal path length. For example, the fifth row
includes all the lattices with a path length of 4, namely the 0-by-4, 1-by-3,
2-by-2, 3-by-1 and 4-by-0 lattices. The corresponding counts of diagonal paths
(and the corresponding entries in Pascal's triangle) are 1, 4, 6, 4 and
1.

Pascal's triangle

This success inspired me to compute a few more paths in the Manhattan
metric. Suppose you are not in a great hurry on your crosstown stroll, and
you do not insist on taking a shortest route but will accept any path that
does not intersect itself--a "self-avoiding walk." With these relaxed
constraints, how many choices do you have? A simple recursive program
quickly found the first few terms of the series: 1, 2, 12, 184, 8512. That
was enough to get a match from the server, which referred me to a diverting
paper by Christoph Dürr of the Laboratoire de Recherche en Informatique at
Orsay, near Paris. The database also gave me the next three terms of the
sequence: 1262816, 575780564 and 789360053252. Who would have thought there
would be so many billions of ways to get from Grand Central Terminal to
Penn Station?

Self-avoiding walks

**A Game of Tag**

Not all visits to the oracle yield such satisfying results. Another
sequence I have submitted to the server comes from a problem first studied
in the 1920s by Emil L. Post. The problem works like this. Take any string
of 0s and 1s at least three digits long, and examine the first digit. If it
is a 0, delete the first three digits and append 00 to the end of the
string; otherwise, delete the first three digits and append 1101. Now
examine the new first digit and repeat the procedure, continuing in this
way as long as possible. Post called the evolving binary string a "tag
system" because the head and the tail of the digit string seem to chase
each other, as in the children's game.

Post's game of tag

A tag system can have three possible fates: The binary string can
dwindle away to nothing, it can grow without limit, or it can enter an
endlessly repeating cycle. One might suppose there would be another
possible outcome, namely that the system could wander forever without
becoming periodic but also without growing beyond some arbitrary length
limit, say *m* digits. But there are only a finite number of binary strings
with no more than *m* digits (How many? See sequence M1599) so that
eventually a string must be repeated. On its second appearance, the
repeated string will produce the same successor it did the first time
around, and this successor will also give rise to the same descendant, and
the evolution of the entire system will thereafter be stuck in a
deterministic loop.

Post was interested in tag systems as a kind of warm-up for the grand
challenge of mathematical logic, as that challenge was perceived in the
early years of the 20th century. It still seemed possible then to devise a
master algorithm for mathematics--a procedure that would mechanistically
decide whether any proposed theorem is true or false. As a preliminary
exercise Post aimed to find a decision procedure for tag systems--a
procedure that would tell whether or not any given starting string would
eventually terminate. Then, perhaps, the same methods could be extended to
the more elaborate formal systems that encode theorems of mathematics. What
happened instead is that Post found even his simple game of tag
intractable, and he made use of this surprising difficulty to show that the
larger ambition of creating a decision algorithm for mathematics cannot
succeed. This was more than 10 years before the work of Kurt Gödel, Alonzo
Church and Alan Turing established the definitive limits of certainty in
mathematics. But Post, who was a young postdoctoral fellow at the time of
his work on tag systems, did not attempt to publish his conclusions until
much later.

As for the tag systems themselves, much remains unknown about them. No
one has found a starting string that does not eventually either dwindle
away or loop, but no one has proved that such strings do not exist. Most of
the known systems enter cycles with a short period. For example, a decade
ago I scanned through about 10,000 tag systems and found periods of 2, 4,
6, 8, 10, 12, 16, 28, 40 and 52. The presence of only even numbers in this
series is easy to explain (think about what happens to the length of the
string on each 00 and 1101 step), but I could see no reason for the process
to favor these particular even numbers and to exclude others.

When I submitted the sequence of known periods to
sequences@research.att.com, the reply was a disappointing "no match." But
Sloane also runs a more sophisticated query service at the e-mail address
superseeker@research.att.com. The superseeker programs work hard to
discover some underlying regularity in a sequence or to transform it into a
known sequence. For example, they try adding small constants to each term
and multiplying the terms by small factors; they look at sums and
differences of adjacent terms; they look at sequences formed by selecting
every second term, and they check the complementary sequence (the sequence
of numbers not in the submitted sequence). Another powerful method of
analysis is to form a generating function, an infinite power series in
which the terms of the sequence appear as coefficients. The
generating-function programs for superseeker were created by Plouffe,
François Bergeron of the Université du Québec à Montréal and Bruno Salvy and Paul Zimmermann of the Institut National de Recherche en Informatique et Automatique in France. Another dozen sequence-analyzing programs, and the 1,200 lines of Unix shell scripts that control the system, were written by Sloane.

I sent the tag-system periods to superseeker. This time the answer did
not come back instantaneously. If the computer was not in fact thinking
deeply about the problem, it nevertheless paused long enough to give that
impression. When the reply finally did show up in my mailbox, superseeker
had a suggestion to offer. Examination of the differences between terms had
revealed a pattern: The last four terms are separated by three equal
intervals of 12, which suggests a very simple rule for continuing the
sequence. As superseeker put it: "Apparently the differences of order 1 in
the difference table of depth 1 have become constant. If this is true then
the next four terms of the sequence are: 64, 76, 88, 100."

The prediction seemed unlikely, but it was provocation enough to send me
back to the computer to try generating more terms. Some 650,000 tag systems
and 50 trillion cpu cycles later, I had a handful of new numbers. None of
the four predicted terms had appeared. Here is the complete set of periods
I have been able to collect so far: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22,
24, 26, 28, 32, 34, 36, 40, 46, 52, 56, 282. What is the meaning of these
numbers? One possibility is that the sequence--if we could see all of its
terms--might turn out to be M0985, the even numbers. Another boring
possibility is that the set of tag-system periods is not a sequence at all
but just a finite and arbitrary collection of numbers. But there remains a
chance that the sequence is infinite and yet includes only a subset of the
even integers, a subset selected by some rule that remains obscure but
might be interesting.

Superseeker's opinion on the question is sensible but not very
illuminating. Given the augmented sequence, it suggests three generating
functions, but what they generate is simply the even numbers. In other
words, superseeker predicts that all the missing terms will eventually show
up. Perhaps that's the best bet. (Later results.)

**Benumbed by Numbers**

What stumps a program like superseeker is not the question too hard to
answer but the answer with too many questions. This is the nature of
inverse problems. If you are asked to evaluate the expression "2 + 2," the
answer is uniquely determined (given certain commonplace assumptions about
the meaning of the symbols). On the other hand, if you are given the answer
"4" and asked what question it comes from, there is no end of valid but
useless responses. Likewise every sequence has an infinity of possible
continuations, although some are more plausible than others. (What is the
next element of this sequence: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39? The answer is 42. They are the numbers of
sequence M0473, values of *n* for which *n*^{2} + *n* + 41 is prime.)

The problem of sifting truth from coincidence gets even worse in the
realm of real numbers. The database of constants for the Inverse Symbolic
Calculator has almost 9 million entries, and Plouffe foresees expanding it
to a billion. It follows that the calculator could assign *some* meaning to
almost any number with fewer than eight or nine digits. Learning that your
telephone number is the solution to a fifth-degree polynomial may be
amusing, but it's unlikely to prove very useful.

The only cure for numerical coincidence is to specify the numbers with
greater precision. In the ISC's tables, numbers are stored with 16 digits
of precision. Thus even in a table with a billion entries, the probability
of finding an exact match by making random probes is only about one in 10 million.

Like the Bell Labs sequence servers, the ISC can apply varying degrees
of sophistication when it searches for the identity of a number. Level 1
does a simple table lookup. If you enter 0.91596559417, it will recognize
Catalan's constant (named for the same Eugène Catalan who is the eponym of
sequence M1459). Level 2, which is not yet implemented, will employ
algorithms based on continued fractions and similar concepts to try to find
a match. Level 3, which is available in a beta-test version, uses another
suite of algorithms to look for linear relations among known constants. For
example, I entered the value 0.453985269150295 at Level 3 and was informed
that these are the first 15 digits of a number equal to

(I didn't just stumble on this number by accident. See Bailey, Borwein
and Plouffe 1995.)

There is no printed version of the Inverse Symbolic Calculator, but
there is a printed precedent. In 1990 Jonathan Borwein and Peter Borwein,
who were then at Dalhousie University in Nova Scotia, published *A
Dictionary of Real Numbers*. This is not a book to keep by your bedside,
although its columns and columns of eight-digit numbers can make it a
valuable reference. ("Wait for the movie," is the usual advice, but in this
case it's "Wait for the Web site.") Plouffe's compilation of real numbers
began independently of the *Dictionary*, but Plouffe and the Borweins have
since made common cause. Jonathan and Peter Borwein now direct the Center
for Experimental and Constructive Mathematics at Simon Fraser University,
where Plouffe has become a research associate.

**Using the Internet Servers**

Compiling tables of mathematical functions was one of the first uses of
automatic computing machinery. No one imagined in the early years that
instead of constructing a table of logarithms, the computer would
ultimately abolish the need for it. And yet the computer also allows us to
build bigger and better tables. What seems to me particularly noteworthy is
that networks of computers now make those tables freely available--through
the generosity of the authors--to an entire worldwide community.

Sloane's sequence servers are accessible to anyone equipped to send mail
to an Internet address. For the basic sequence server, which merely looks
the sequence up in the database, the address is sequences@research.att.com.
The body of the message should include at least one line of the following
form

lookup 2 4 6 8 10 12 14 16 18 20 22

Note that the terms of the sequence are separated by spaces rather than
commas. As many as five lookup requests can be included in a single
message. A message without lookup requests elicits a help file.

Requests to the superseeker have the same format, but a message must
have only one lookup line, and furthermore only one request per hour is
allowed. (Running the superseeker programs puts a significant load on
Sloane's computer.) The address is superseeker@research.att.com.

Both addresses can be reached through Sloane's home page on the World Wide Web. This site also provides copies of files listing corrections and recent additions to the database.
The Inverse Symbolic Calculator operates exclusively via the Web. Instructions for using the
three versions of the program are given at the site.
A related Web site, with extensive expository articles on mathematical
constants, is run by Steven Finch of Mathsoft, Inc.

**Bibliography**
Bailey, David, Peter Borwein and Simon Plouffe. 1995. "On the Rapid
Computation of Various Polylogarithmic Constants." Submitted.

Borwein, Jonathan, and Peter Borwein. 1990. *A Dictionary of Real
Numbers*. Pacific Grove, Calif.: Wadsworth & Brooks/Cole.

Dürr, Christoph. 1994. "Deux Mariages et aucun Enterrement."

Hayes, Brian. 1986. "Tag--You're It." *Computer Language*, August,
pp. 21-28.

Post, Emil L. 1941. "Absolutely Unsolvable Problems and Relatively
Undecidable Propositions: Account of an Anticipation." In *Solvability,
Provability, Definability: The Collected Works of Emil L. Post*, ed.
Martin Davis. 1994. Boston: Birkhäuser.

Sloane, N. J. A. 1973. *A Handbook of Integer Sequences*. New York:
Academic Press.

Sloane, N. J. A. 1994. "An On-Line Version of the Encyclopedia of
Integer Sequences." *The Electronic Journal of Combinatorics*. Vol. 1, Feature F1.

Sloane, N. J. A., and Simon Plouffe. 1995. *The Encyclopedia of
Integer Sequences*. San Diego: Academic Press.

© Brian Hayes

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