Download Full text

FIBONACCI NUMBERS OBTAINED FROM PASCAL'S TRIANGLE
WITH GENERALIZATIONS
H. C. WILLIAMS
University of Manitoba, Winnepeg 19, Canada
1.
INTRODUCTION
Consider the following a r r a y of numbers obtained from the first
k lines of P a s c a l ' s
Triangle,
1
0
0
1
1
0
1
2
.1
0
...
0
0
C-1) (v)
2k
4k
If we let the element in the i + 1
column and n
0, 1, 2, ••• , k - 1) and
F.
12 F i,n-j
row be F .
M,
i,n'
then Fr . i ,„ n=" (( "i ) ) (n,i =
G = 0, 1, 2, •••, k - 1; |n| = k, k + 1, •••)
1=1
If k = 2.5 F~
= f , - , F= f , where f is the n
Fibonacci number; and if k = J3,
0,n
n+1
l,n
n
n
F„
= L . and F r t
= K - , where L and K a r e the general Fibonacci numbers of
l,n
n+1
2,n
n-1
n
n
WaddiU. and Sacks [8], Also, F f e _ 1 h = fh k where the fQ fc a r e the k-generalized Fibonacci numbers of Miles [ 5 ] , and F A
U.
= U,
of Ferguson [2]. Both the numbers f
. and
a r e of use in polyphase merge sorting techniques (see, for example, Gilstad [3] and
rL9 n
Reynolds [6]).
The purpose of this paper is to investigate some of the properties of a more general
set of functions which include the functions F .
(i = 0, 1, 2, • • • , k - 1 ) and several others
is n
as special c a s e s .
405
406
FIBONACCI NUMBERS OBTAINED FROM PASCAL'S
[Oct.
2. NOTATION AND DEFINITIONS
Let {c^, a2i * • • , % } be a fixed set of k integers such that
k
-nw \
k - 1 _*_
F(x) = x
has distinct zeros
- a^x
k-2
+ a2x
p 0 , Pi, • • • , Pk_-p
,
, / i\k
+ ...
+ (-1) ^
Let a 0 , a l9 • • • , a, - b e a n y k integers and define
k-1
p
=
i
l
Jl *A
d~j
j=o
l
(i = 0, 1, 2, •••, k - 1)
l
Finally, let
D =
1
Po
Pi
...
Po
1
Pi
Pi
2
...
Pi
k-l
P2
p
k-l
1
p
...
P
k-1
A = |D|.
We shall concern ourselves with the functions
(2.1)
A.
i,n
1
Po
•
p}- 1
00
i+1
Po
•
Po
1
Pi
•
k-1
Pi
01
i+1
Pi
•
k-1
Pi
i-1
k-l
*k-l
1
p
k-l
• '•
p
p
i+1
k-l
'"
p
k-lj
k-l
(i = 0, 1, 2, • • • , k - 1) .
It i s c l e a r that
k-1
A
(2.2)
i,n
=
f EV?
(i =
0,1, 2 , . - . k - l ) ,
i=o
where c .
is the cofactor of p] in D.
If &i = 1, a. = 0 (i = 0, 2, 3, • • • , k - 1), we have <p. = p. ; and, in this c a s e , we
define A.
to be z.
. These functions, which a r e quite useful in the determination of the
1972]
TRIANGLE WITH GENERALIZATIONS
properties of A.
407
, have been dealt with in some detail by authors such as Bell [1] 9 Ward
[9] a n d S e l m e r [ 7 ] , When a = (-1) 1
(i = 1, 2, ••• , k), {z. -
} is the general F i b -
Ki-x 9 n
i
onacci sequence discussed by Miles [5] and Williams [ l 0 ] .
Since matrix methods a r e advantageous in the treatment of the A.
functions, we in-
i9n
troduce the following:
C =
"00
"10
"k-1 0
"01
"11
"k-1 1
l k-1
k-1 k-l_
c
C
0 k-1
C. = diag (c. o 9 c . r
' c i k-1 ) ,
i,k-l
°1.0
Z
Z. =
l
i,k-l
n,r
i,2
i,k
%2k-2
i,k
h
01
,n+r
01
"k-1
,n+r
,n+(k-l)r
^n+fe-Dr
,n+(k-l)r
k-l,n
0,n
B
A
^0 s n+r
n9r
A
A
09n+(k-l)r
k-1,n+r
l9n+(k-l)r
k-l,n+(k-l)r
A.
i,n
A-i 9 n+r
^
B.
i,nsr
A
i9n+(k-l)r
3.
The A.
l9n+r
A.
i,n+2r
A.
i 9 n+(k-l)r
A.i,n+kr
A
A
i,n+kr
i,n+(2k-2)r
SPECIAL CASES
functions include a number of interesting functions as special c a s e s .
have already mentioned the z.
We
functions in the previous section and in this section we d e -
scribe several other special c a s e s . We first show the relation of the function F .
to A.
i,n
i,n
.th
Let H. (j = 1, 2, . . . 9 k) be the j
elementary symmetric function of 0O9 0 j ,
4>k_l9
then
408
FIBONACCI NUMBERS OBTAINED FROM PASCAL'S
[Oct.
k
+1
(3.1)
( 1 ) J +T
1 H A
=X)W
"
J i>n-j
3=1
A. j n
If
•'[(•)-(•->)]
«i = ( - i r i m - L , 1 1
and a0 = aA = 1,
a = i, 2. •••,k)
we have 0. = 1 + p. and H. = (-1) J + .
a. = 0 (I = 2, 3, • • • , k - 1),
Hence,
k
=y^A.
A.
i,n
L-d
3=1
and
V
i,n
=
(0 <
(i)
i,n-j
i,
n <
k - 1)
thus, in this c a s e ,
A.
= F.
i,n
i,n
When k - 2, at = 1 - 2a, a2 = a2 - a - 1, SLQ = a,
-1,
and AQ
n
= afn + f ^ ,
A.
Q
= fn«
If ^
An
= 2n_1£
0,n
n
where 9- is the n
Lucas number.
If
=0,
and
aA = 1, we have
5
a2 = " >
a
Hi = 1, H2 =
a
o = i = 1> we have
A,
= 2n"1f ,
l,n
n
(x l9 yA) i s the fundamental solution of the Pell
equation
x2 - dy2 = 1 ,
(3.2)
ax = 0,
a2 = -d,
a 0 = xl9
at = yi ,
Then AA
= x and A,
= Jy , where (x , Jy ) is the n
solution of (3.2).
0,n
n
l,n
n
n n
When k = 3, we also have some interesting c a s e s . For example, if a± = a2 = a% =
2 and a0u = -ai1 = aL2 = 1, we have Hi = -H 2 = H63 = 1 and AA
= UQ , A.
= -L . ,
*
0,n
3,n
l,n
n-1
AQ&, n = ^Q
j_i
=
K
.
If
(x
yi,
z
)
is
a
fundamental
solution
of
the
diophantine
equation
l9
A
o, n+i
n
(Mathews [4])
(3.3)
x 3 + dy 3 + d 2 z 3 - 3dxyz = 1 ,
1972]
TRIANGLE WITH GENERALIZATIONS
and a0 = a2 = 0, az = d,
a0 = x l s
a! = y l s
a2 = z l s
409
then all the solutions of (3.3) are
given by
(A
0,n'
A
l,n>
A
2sn)
4.
(|n
l
=
0,1,2,...).
IDENTITIES
We now obtain several of the important relations satisfied by the A.
functions.
It
will be seen that each of these relations is a generalization of a corresponding identity s a t isfied by the Fibonacci numbers.
The most important properties of the Fibonacci numbers
a r e the identities which connect the numbers f , , f
and f
to other Fibonacci num—
n+m n - m
nm
b e r s . F o r the sake of convenience, we shall call these relations the addition, subtraction,
and multiplication formulas.
By (2.1),
where we denote by Bf
B. __,
= P
C.P f
i,n+m,r
n , r I m, r
the transpose of the matrix B. Since
CD = DC = I ,
B. ^
= P
C'D'C.DCP'
= B
Z.B!
;
i,n+m,r
n,r
I
m,r
n,r I m,r
hence,
A.
, = (AA A, AQ
« . . A.
)Z.(A
A,
. . . A.
)?
i,m+n
0,n l , n 2,n
k,n l A0,m l , m
k,m
(4.1)
k-1 k-1
ZLJ z L r Z i s h + j
h=0 j=0
h,n
j,m
This is the addition formula for A. >n*
ByJ the definition of A.
it follows that
i,n
k-1
(4.2)
3=o
thus,
k-1
h ,m
p. 0.
r P ?*A.
3=o
k-1
/ k-1
]=o
\ j=o
=
410
[Oct.
FIBONACCI NUMBERS OBTAINED FROM PASCAL'S
Now if H = H k = 0 o 040i ••• <%_!>
H m A.
i,n-m
A
00
Po0o
i-1 ,m
Po 0o
,n
0o
i+l^ni
Po 0o
k-1 ,m
Po 0o
,m
0i
,m
pi(p1
i-1 ,m
Pi 0i
,n
0i
i-l^m
Pi 0i
k - 1 ,m
Pi 0i
i+1 , m
k-l.m
,m
n
,m
i-1 ,m
,m
By (4.2)
H m A.
i,n-m
(4.3)
k-1
y ^ -A.
*-J
3=0
0,]
j,i
2wZ0si+j-lAj,m
A
0,m
Z/o,i+j+lAj,m
'*'
Z/o,i+k-lAj, m
Z)Zl,i+j-lAj,m
A
l,m
Szl.i+|+lAj,m
'"
2 Z l , i + k - l A jj ,, m
5-XjAj,m
"'
2JZk-l»JAJ»m
""X/Zk-l.i+3-lAJ»ni-
this is the subtraction formula for A.
A
k-l,mzJZk-l,i+j+lAj,m
' "SZk-l,i+k-lAj,m
.
i,n
Since
k-1
1 \ ^
A.
i,nm
^nm
1=0
A.
i,nm
k
/k-i
\
j=0
\h=0
/
]
F r o m (4.2), we get the multiplication formula
k
i,nm
T
Z-/ii-i2 - ••• ik- j = i
where the sum is taken over all non-negative integers
and £ ( j - l ) i . .
i.
z.
J> m i, s
ii, i 2 , ••" , % such t h a t ^ i .
= m;
1972]
TRIANGLE WITH GENERALIZATIONS
We may easily evaluate the determinants of B
n, r
411
, and B.
by first introducing
i;ii) r
the matrix.
,n.
Q.
& (k-l)n
s (k-l)n
^(k-l)n
k-1
Now
0,0
k-l90
l9n
Ss-l.n
l,(k-l)n
Vl,(k-l)n
0,n
QnC =
A
A
0,(k-l)n
hence
lsn
k,n
l,2n
^k,2n
l,(k-l)n
^ k , (k-l)n
K\
Since
iPn,r| =
H
K
we have
(4.5)
B
n,r
n-1 9 r
l,r
^l,2r
^n-l,2r
%(k-l)r
nrl,(k-l)r
A
and
A
(4.6)
B.
i9n,r
H,2r
Hn|Z.|
n-l9r
l9r
A
.n-l,2r
i
A
l,(k-l)r
An - l ( k - l ) r
9
REFERENCES
1. E. T. Bell,
n
Notes on Recurring Series of the Third O r d e r , " Tohoku Math. Journal,
Vol. 24 (1924), pp. 168-184.
2. David E. Ferguson, "An Expression for Generalized Fibonacci N u m b e r s , "
Fibonacci
Quarterly, Vol. 4 (1966), pp. 270-273.
3. R. L. Gistad, "Polyphase Merge Sorting — An Advanced Technique," P r o c . Eastern Joint
Computer Conference, Dec. I960.
4. C. B. Mathews, "On the Arithmetic Theory of the F o r m x 3 + ny 3 + n 2 z 3 - 3nxyz," P r o c .
London Math. Soc. , Vol. 21 (1891), pp. 280-287.
5. E. P . Miles, J r . , "Generalized Fibonacci Numbers and Associated M a t r i c e s , " Amer.
Math. Monthly, Vol. 67 (I960), pp. 745-752.
FIBONACCI NUMBERS OBTAINED FROM PASCAL'S
TRIANGLE WITH GENERALIZATIONS
412
iq?
6. S. W. Reynolds, "A Generalized Polyphase Merge Algorithm," Communications of the
ACM, Vol. 4 (1961), pp. 347-349.
7. E. S. Seimer, Linear Recurrence Relations over Finite Fields, University of Bergen,
Norway, 1966.
8.
M. E. WaddiU and L. Sacks, "Another Generalized Fibonacci Sequence," Fibonacci
Quarterly, Vol. 5 (1967), pp. 209-222.
9. Morgan Ward, "The Algebra of Recurring S e r i e s , " Annals of Math. , Vol. 32 (1931),
pp. 1-9.
10. H. C. Williams, "On a Generalization of the Fibonacci N u m b e r s , " P r o c . Louisiana
Conference on Combinatorics, Graph Theory and Computing, Louisiana State University,
Baton Rouge, March 1-5 1970, pp. 340-356.
[Continued from page 401. ]
REFERENCES
1. Brother U. Alfred, "Exploring Special Fibonacci Relations," Fibonacci Quarterly, Vol.
4, Oct. , 1966, pp. 262-263.
2. D. Zeitlin, "Power Identities for Sequences Defined by W n + 2 =
dw
n+1
- c W n , " Fibon-
acci Quarterly, Vol. 3, Dec. 1965, pp. 241-256.
[Continued from page 404. ]
REFERENCES
1. A. F. Horadam, American Math. Monthly, Vol. 68 (1961), pp. 751-753.
2. C. W. Trigg,
Mathematical Quickies, McGraw Hill (1967), pp. 69, 198.
3. E. M. Cohn, submitted to Fibonacci Quarterly.
4. A. F . Horadam, American Math. Monthly, Vol. 68 (1961), pp. 455-459.
•"ANNOUNCEMENT.*.
The Editor (who always parks in a Fibonacci-numbered parking space) noted the following in the latest publication of the Fibonacci Association, A P r i m e r for the Fibonacci Numb e r s : There a r e 13 authors, each of whom wrote a Fibonacci number of a r t i c l e s .
Each c o -
author has a Fibonacci number of articles with a given co-author. There a r e 11 a r t i c l e s with
one author, and 13 articles have co-authors.
Of the twenty-four a r t i c l e s , 13 a r e P r i m e r
a r t i c l e s , and 11 a r e not.
The P r i m e r , co-edited by Marjorie Bicknell and V. E. Hoggatt, J r . , is a compilation
of elementary articles which have appeared over the y e a r s . These articles were selected and
edited to give the reader a comprehensive introduction to the study of Fibonacci sequences
and related topics.
$5.00.
The 175-page P r i m e r will be available in the Fall of 1972 at a cost of