
Fibonacci Forgeries
MATHEMATICAL RECREATIONS
by Ian Stewart
Scientific American, May 1995
A toast, ladies and gentlemen: To the Queen!” We stood, dutifully raised
our glasses and murmured the sovereign’s name. It should have been
a poignant moment, but it was spoiled by the person next to me, who flopped
back into his chair and muttered, “Thank God, now I can smoke!” (It
is a quaint British custom that at a formal dinner no one may smoke before
the party drinks to the queen’s health.)
“I’d rather you didn’t,” I said. “I’m a
nonsmoker.”
“You had the smoked salmon,” he said and roared with laughter as
he lit a cigarette. I wrinkled my nose and glanced at his badge: Richard Byrd.
He had penciled in, “Call me Dicky.”
“The salmon, by the way, was terrible,” I told him. “In fact,
I think it was really dogfish dyed orange.” He did not look surprised.
The annual dinner of CATDOG (the Charitable Association for TaxDeductible
Offerings of Generosity) was always a disaster.
“I wondered why the fish was wearing a flea collar,” said the lady
on my left. I had met her at other gatherings: Amanda BanderGander, a leading
light of the local Animal Protection Society.
“No, dear, you just dropped your napkin ring on it,” her husband
yelled from five places down on the opposite side of the table. Alexander
BanderGander, a lawyer, was sandwiched between Athanasius Fell, a doctor,
and Dennis Racket, an old friend of mine from the tennis club.
“So what business are you in?” Dicky asked me.
I leaned closer to him and whispered, “I’m a mathematician.”
As usual, though I had spoken below the threshold of aural perception, it
was a partystopper. The entire table went silent.
“I was never—” he began.
“Any good at math at school,” I finished. “That’s what
they all say.”
“Hated it then,” Athanasius mumbled. “Still do.”
“I must be the exception!” a voice boomed at my right. “Absolutely
loved it! Name’s Adam Smasher. I’m a nuclear physicist. I have
a little puzzle I’ll ask all of you. What’s the next number in
the sequence 1, 1, 2, 3, 5, 8, 13, 21?”
“Nineteen,” I grunted automatically, while battling with a bread
roll seemingly baked with cement.
“You’re not supposed to answer,” he said. “Anyway,
you’re wrong—it’s 34. What made you think it was 19?”
I drained my glass. “According to Carl E. Linderholm’s great
classic Mathematics Made Difficult, the next term is always 19, whatever
the sequence: 1, 2, 3, 4, 5—19 and 1, 2, 4, 8, 16, 32—19. Even
2, 3, 5, 7, 11, 13, 17—19.”
“That’s ridiculous.”
“No, it’s simple and general and universally applicable and thus
superior to any other solution. The Laplace interpolation formula can fit
a polynomial to any sequence whatsoever, so you can choose whichever number
you want to come next, having a perfectly valid reason. For simplicity, you always
choose the same number.”
“Why 19?” Dennis asked.
“It’s supposed to be one more than your favorite number,” I
said, “to fool anyone present who likes to psychoanalyze people based
on their favorite number.”
“Nonsense. I’ll tell you the real answer,” Adam said. “Each
number is the sum of the previous two. So the next is 13+21, or 34, then 55,
then 89, then 144 and so on. It’s the—”
“Fibonacci sequence,” I interrupted. “God, I’m so fed
up with the blasted Fibonacci sequence! Even the name’s phony! ‘Leonardo
Fibonacci, son of Bonacci!’ That’s a nickname invented by Guillaume
Libri in 1838, long after Fibonacci died. The famed Fibonacci was in fact named
Leonardo Pisano Bigollo. Pisano means that he lived in Pisa; no one knows what
Bigollo means. At any rate, his sequence ought to be called the Leonardo Pisano
Bigollo sequence, except that’s too long.”
Fibonacci or Forgery?
Sequence 1 is 2, 3, 5, 8, 13 ... a_{n}, where
a_{n} equals 1 plus the sum of the first n terms of the Fibonacci sequence.
For example, a_{1} = 1 + 1 = 2; a_{2} = 1 + (1 + 1) = 3; a_{3} = 1 + (1 +
1 + 2) = 5 and so on.
Sequence 2 is 1, 3, 8, 21, 55 ... a_{n}, where
a_{n} equals the sum of the first n terms of the sequence formed by
removing every other Fibonacci number. So a_{1}=1; a_{2}=1+2=3; a_{3} = 1
+ 2 + 5 = 8 and so on.
Sequence 3 is 1, 1, 2, 3, 5, 8 ... a_{n},
where a_{n} is the number of ways in which you can arrange n coins
in horizontal rows such that all the coins in each row touch and
every coin above the bottom row touches two coins in the row below
it.
Sequence 4 is 1, 2, 5, 13 ... a_{n}. It is
the same as sequence 3, except that now n is the number of coins
in the bottom row.
Sequence 5 is 1, 2, 5, 13 ... a_{n}, where
a_{n}=(n – 1)2^{n–2}
+ 1.
Sequence 6 is 1, 2, 5, 13 ... a_{n}, as defined
by the number of disconnected graphs with n + 1 vertices.
Sequence 7 is 1, 1, 2, 3, 5, 8 ... a_{n},
where a_{n}equals the integer nearest to ((1 + 5)/2)^{n}/ 5.
Sequence 8 is 1, 2, 5, 13 ...a_{n}, defined
by the number of connected graphs with n + 2 vertices having just
one cycle.
Sequence 9 is 1, 2, 5, 13 ... a_{n}, where
an is the coefficient of xn+2/(n + 2)! in the power series solution
to the differential equation y¢¢ = exy, which starts y
= 1 + x ^{2}/2!+ x ^{3}/3! + 2 x ^{4}/4! + 5 x ^{5}/5!
+ 13 x ^{6}/6!...
Sequence 10 is 1, 2, 5, 13...a_{n}, where
an equals a_{n–1}+na_{n}_{–2}
, when a_{1}=a_{0}=1/2.
Sequence 11 is 2, 3, 5, 8, 13...a_{n}. In
this sequence, apartment blocks of n floors are to be painted blue
and yellow, with the rule that no two adjacent floors can be blue.
(They can, however, be yellow.) Let a_{n} be the number of ways to
paint a block with n floors.

“You mathematicians have a lot of attitude,” Dicky
remarked. “At least, an attitude that differs from most other people’s.”
“Proof,” Adam declared. “Mathematicians always want to prove
things. That’s certainly a strange attitude. Never understood why myself.
If you keep trying it, and it keeps working, it’s got to be right! So
why waste time getting into all sorts of logical tangles proving the silly
thing?”
“Well, why do you physicists bother doing experiments? If a theory tells
you what you want to hear, why not just assume it’s true?”
“Because you can’t just go around believing theories without testing
them!”
“And mathematicians don’t think you can go around believing theorems
without proving them. Alexander, why do lawyers insist on cases being tried
in court? Why not just let the judge look at the evidence and decide whether
the defendant has committed the crime?”
“You can’t do that! There could be a miscarriage of justice!”
“Right. That’s what mathematicians worry about when they insist
on proofs. They don’t want to find out later they were wrong. It might
be embarrassing.”
Adam shook his head sadly. “You know full well it’s not like
that. Mathematics is basically simple. If you can see an obvious pattern,
it can’t be coincidence. Why bother to prove it?”
I thought for a few seconds. “I’ll give you an example. Here’s
a sequence, and I want you to tell me the next number.”
“I’ll do my best.”
“1, 1, 2, 3, 5, 8, 13, 21, 34, 55,” I said.
He looked puzzled. “Don’t be silly. I just asked you that one.
It’s the Fibonacci sequence.”
“Is it? Then what’s next?”
“89,” Adam replied.
“Wrong. It’s 91.”
“But it looks just like the—”
“You’re leaping to conclusions, Adam, based on previous prejudices.
Most injudicious. Your sequence was the Fibonacci sequence, but the nth term
in my sequence is the least integer not less than Öe n–2, where Öe
= 2.71828, the base of natural logarithms. I want the 11th term, which is the
least integer not less than Öe9, or 90.017. That’s 91.”
“Humph. Well, that’s an accident, a rare exception. I’ll
believe you if you can show me some more examples of misleading sequences,” Alexander
interjected. “Can you? Or have you exhausted your repertoire?”
“There are hundreds,” I said. “It’s easier than you
think. Richard K. Guy, a mathematician at the University of Calgary, collects
them. He refers to them all as the Strong Law of Small Numbers. There aren’t
enough small numbers to meet the many demands placed on them, so what look
like patterns involving small numbers might just be coincidence. And often
are.”
I showed them 11 sequences that looked like the Fibonacci numbers, or the
alternate Fibonacci numbers 1, 2, 5, 13 and so on (see box above). I asked them to decide
which were really Fibonacci sequences and which were Fibonacci forgeries (see box below for answers).
My companions began to argue about whose solution was right when I proposed
one last puzzle. “What is the next term in the sequence 3, 5, 7, 11,
13, 17, 19...”—I carried on for some time listing the odd primes—“...331,
337?”
“Those are the odd primes,” Adam said. “You can’t generate
that many primes by accident. The next term must be—uh—347.”
“Are you sure?” I asked quietly.
Solutions to the Misleading Sequences
Sequence 1: Fibonacci. For example, to
form a_{6}, you add 8, the sixth term in the Fibonacci sequence. The
sum is 8+13, a sum of consecutive Fibonacci numbers. By definition,
then, the answer, 21, is itself a Fibonacci number. This pattern
continues.
Sequence 2: Fibonacci. For example, to
go from a_{5} to a_{6}, you add 89. The sum is 55+89, a sum of consecutive
Fibonacci numbers. By definition, then, the answer, 144, is itself
a Fibonacci number. This pattern continues.
Sequence 3: Forgery. The sequence continues
12, 18, 26.
Sequence 4: Fibonacci. The proof depends on the
identity f_{2n–1} = f_{2n–3} + 2f_{2n–5} + 3f_{2n–7}
+ ... + (n–1) f_{1} + 1 for Fibonacci numbers f_{n}.
Sequence 5: Forgery. The sequence continues
33, 81, 193.
Sequence 6: Forgery. The sequence continues
44, 191, 1,229, 13,588.
Sequence 7: Fibonacci. This pattern follows
from Binet’s formula f_n = [((1 + \sqrt{5})/2)^n + ((1  \sqrt{5})/2)^n]/ \sqrt{5}.
Sequence 8: Forgery. The sequence continues
33, 89, 240, 657, 1,806.
Sequence 9: Forgery. The sequence continues
36, 109, 359, 1,266.
Sequence 10: Forgery. The sequence continues
38, 116, 382.
Sequence 11: Fibonacci. Consider, for example,
a fivefloor building. The fifth floor can be either yellow or blue.
If it is yellow, the rest of the building can be painted in any possible
way for a fourfloor building. If it is blue, the fourth floor must
be yellow, and the rest of the building can be painted in any possible
way for a threefloor building. So a_{5}=a_{4}+a_{3}, as it is for Fibonacci
numbers. The pattern is general.
Odd Prime Sequence: Forgery. The sequence
3, 5, 7, 11...331, 337 and so on consists of all the numbers n that
divide 2^{n–1}–1 exactly. The next term
is 341, which is not prime (11 x 31=341) but divides 2^{340}–1. 
FURTHER READING
MATHEMATICAL GAMES: PATTERNS IN PRIMES ARE A CLUE TO THE STRONG LAW
OF SMALL NUMBERS. Martin Gardner in Scientific American, Vol. 243,
No. 6, pages 18–28; December 1980.
A DIARY ON INFORMATION THEORY. Alfréd Rényi. Wiley, 1987.
THE STRONG LAW OF SMALL NUMBERS. Richard K. Guy in American Mathematical
Monthly, Vol. 95, No. 8, pages 697– 712; October 1988.
THE SECOND STRONG LAW OF SMALL NUMBERS. Richard K. Guy in Mathematics
Magazine, Vol. 63, No. 1, pages 3–20; February 1990.

