ON A CLASS OF NONLINEAR BINOMIAL SUMS
D. A. Lind
U n i v e r s i t y of V i r g i n i a , C h a r l o t t e s v i l l e ,
Virginia
It is known ( [ 2] ; [4] ; [5 ] ) that the Fibonacci n u m b e r s m a y b e
formed by adding the binomial t e r m s on diagonals of P a s c a l ' s T r i angle. Recently in this Q u a r t e r l y V. C. H a r r i s and Carolyn C. Styles
[ 3] g e n e r a l i z e d the Fibonacci sequence by extending t h e i r c o n s i d e r a tion to s u m s along s t r a i g h t diagonals of any positive " s l o p e " o r i g i n a t ing in the f i r s t column.
As they noted, those s u m s a r e s p e c i a l c a s e s
of the linear binomial s u m s investigated by Dickinson [l] .
H e r e we
c o n s i d e r a nonlinear g e n e r a l i z a t i o n of this connection, in which each
sum contains a "dogleg" of binomial t e r m s .
We then note that t h e s e
s u m s obey the s a m e difference equation as the binomial coefficients.
F r o m this equation and a set of a u x i l i a r y n u m b e r s we d e r i v e some
a r i t h m e t i c p r o p e r t i e s , including connections with the Fibonacci n u m bers,
and develop some g e n e r a l r e c u r r e n c e s .
Because of this close
connection with the binomial coefficients, it is not s u r p r i s i n g that m o s t
of the p r o p e r t i e s g i v e n h e r e s t e m from c o r r e s p o n d i n g p r o p e r t i e s of the
binomial coefficients.
We define
L(n, r), the r - t h o r d e r nonlinear binomial sum, as
the sum of the f i r s t
r t e r m s of the ( n - l ) - t h row of P a s c a l ' s Triangle
plus the t e r m s on the r i s i n g s t a i r s t e p diagonal originating at the r - t h
term.
Thus
[^]
r - 1
(1)
Mn..r)=
2
i=0
n
1
( [ )+
LA
2
j=l
(££}),
3
w h e r e | ] denotes g r e a t e s t i n t e g e r , and the r i g h t - m o s t s u m s is z e r o
if[;2il]<l.
The s u m s L(n, 1) = L ( n - 1 , 2 ) = F , the n - t h Fibonacci
n u m b e r , a r e those p r e v i o u s l y c o n s i d e r e d in [2] , [4] , and [5] . F o r
r = 3 we obtain the following s e r i e s .
292
293
ON A CLASS OF NONLINEAR BINOMIAL SUMS
Dec.
1
Thus
is
L(l, 3) = 1, L(2, 3) = 2, L(3, 3) = 4, etc.
1, 2, 4, 8, 15, 27, 47, 80, 134, • • • .
The 4-th order sequence
'
The connection between the Fibonacci numbers and binomial coefficients previously mentioned may be written as
1—1
n
I
i=0
The difference
between the nonlinear binomial sums and Fibonacci
numbers is therefore
r-1
F
n+r-l "
L { n ? r )
nTT
=
S
(
i
.
r-1
) -
2 (
i=0
i=0
which is a polynomial in n of degree
r-3
for
r L. 3.
),
By evaluating
the right side of this equation for small values of r, we find, in addition to
(2a)
(2b)
L(n, 1) = L(n-1, 2) = F , that
L
* n ' 3>
= F
n+2 - l >
L(n, 4) = F n + 3 - n - 1
1965
ON A CLASS OF NONLINEAR BINOMIAL SUMS
294
Also, since
n-1
s
,
.
(n:l)-zn-1
,
i=0
we see from the definition that
L(n, r) = 2n~l
Let the difference o p e r a t o r
f(n).
(n < r) .
A
be defined by A f( n )
=
f( n +l) -
Then the r e c u r r e n c e r e l a t i o n for the binomial coefficients may-
be r e p r e s e n t e d as
(3)
J
(? =
^n^ .
F r o m this and the explicit r e p r e s e n t a t i o n in (1), the i m p o r t a n t difference equation follows that
(4)
A L(n, r) = L(n, r - 1 )
Defining the i t e r a t e d difference o p e r a t o r
Ak
AK
VI
f( n )
=
k_1
A
f( n )| for
n [A
n
Ar"2
V
A "
n
1
.
^
by A
n
f(n) =
A
f(n),
n
k > 1, it is of i n t e r e s t to note that
L ( n , r ) = L(n, 2) = F n + 1 ,
L(n, r) = L(n, 1) = F
n
.
It is a p p a r e n t that (4) is indeed the s a m e difference equation satisfied
by the binomial coefficients in (3), the only change being in the initial
values.
Thus by using (4) and the e a s i l y d e t e r m i n e d boundary conditions
L(n, 1) - F n , L ( l , r ) = 1 ,
we m a y c o n s t r u c t a table of
L(n, r) in which each t e r m is the sum of
the t e r m above it and the t e r m above and to the left.
Since the sequence
L(n, 2) = F
,
s a t i s f i e s the r e c u r r e n c e
relation
L(n+2,2) = L(n+1,2) + L(n, 2) ,
^95
ON A CLASS OF NONLINEAR BINOMIAL SUMS
we take the a n t i - d i f f e r e n c e of this
r-2
Dec.
t i m e s and obtain the. g e n e r a l
r e c u r r e n c e r e l a t i o n for r - t h o r d e r t e r m s as
(5)
L(n+2, r) = L(n+1, r) + L(n, r) + A(n, r) ,
Table of
L(n, r)
\
r - 1
2
3
4
5
6
7
8
1
1
1
1
1
1
1
1
1
1
1
2
2
1
2
4
4
2
4
2
3
2
4
2
2
2
4
4
4
2
4
8
8
8
8
8
8
8
5
5
8
7
15
16
16
16
16
16
16
8
13
12
20
27
31
32
32
32
32
32
13
21
33
47
58
63
64
64
64
64
8
21
34
54
80
105
121
127
128
128
128
9
34
55
88
134
185
226
248
255
256
256
55
89
143
222
319
411
474
503
511
512
n
2
i •
3 i
4 1
5 |
6
7
10
1
1
•
3
w h e r e the a u x i l i a r y n u m b e r s
_JL-
10
A(n, r) obey
A r " 2 A(n, r) = 0
n
with the initial conditions
A(n, 1) = A(n, 2) = 0
(n > 1); A ( l , r) = 1
(r > 3)
.
T h e s e n u m b e r s a l s o obey the binomial r e c u r r e n c e
A A ( n , r ) = A(n, r - 1 ) ,
so that we m a y e a s i l y c o n s t r u c t a table of A(n, r) from the initial conditions using the s a m e rule of f o r m a t i o n as that for
from this table that while
L(n, 1) and
L(n, r ) .
It a p p e a r s
L(n, 2) a r e sequence of the F i b -
onacci type, the next two obey the slightly m o r e c o m p l i c a t e d r e c u r r e n c e s
L(n+2, 3) = L(n+1, 3) + L(n, 3) + 1 ,
L(n+2, 4) = L(n+1, 4) + L(n, 4) + n .
1965
296
ON A CLASS OF NONLINEAR BINOMIAL SUMS
These a r e r e a d i l y proved by using equations (2a) and (2b).
Table of A(n, r)
10
n
0
0
1
1
1
1
1
1
1
1
0
0
]
2
2
2
2
2
2
2
3
0
0
1
3
4
4
4
4
4
4
4
0
0
1
4
7
8
8
8
8
8
5
0
0
]
5
11
15
16
16
16
16
6
0
0
3
6
16
26
31
32
32
32
7
8
0
0
3
7
22
42
57
63
64
64
°
0
3
8
127
128
0
3
93
o
0
3
9
10
99
163
120
0
29
37
64
46
140
256
219
382
247
466
255
502
1
2
9
10
;
l
1
We m a y e s t a b l i s h from (4) and (5) that the r e c u r r e n c e formula
with r e s p e c t to r
is
L(n, r) = L(n, r - 1 ) + L(n, r - 2 ) - A(n, r) .
F r o m this, with (4) again, it follows that
A(n, r) = L(n+1, r - 1 ) - L(n, r) .
This l a s t equation m a y be u s e d to e s t a b l i s h that
L(n, r) +
r-1
Z A(n+i, r - 1 ) = F
n+r-1
i=0
Taking n = 1, we see that the slant sums of the A(n, r) a r e Fibonacci
n u m b e r s diminished by a unity, i. e.
2
i=l
A(i, r-i+1) = F
- 1 .
I t i s a l s o i n t e r e s t i n g to note that the A(n, r) obey the c u r i o u s diagonal
recurrence
297
- ON A CLASS OF NONLINEAR BINOMIAL SUMS
Dec.
A(n+1, r + 1) = 2 A(n, r) + (^~*) .
The r e c u r r e n c e (5) m a y be e a s i l y extended by induction to
k
L(n, r) = F k + 1 L(n~k, r) + F R L ( n - k - l , r) + S F . A ( n - i - l , r) (0 < k < n),
i=l
and the analogous extended r e c u r r e n c e with r e s p e c t to r
is
L(n, r) = F k + 1 L(n, r - k ) + F k L(n, r - k - 1 ) - 2 F . A(n, r-i+1) (0 < k <n).
i=l
We r e m a r k that setting
r = 1 in the f o r m e r r e c u r r e n c e
gives
the
f a m i l i a r F i b o n a c c i identity
F
= F
F
+F F
n
k+1 n - k
k n-k-1
We m a y p r o v e by induction that for r > 1
k
2
i=l
k
2
L(i, r) = L(k+1, r+1) - 1 ,
A(i, r) = A(k+1, r+1) - 1 ,
i=l
.
•
•
which t o g e t h e r imply
n
Z
i=l
[L(i, r) + A(i, r)] = L(n+2, r ) - 2 (r > 1) .
Finally, we extend the definition of L(n, r) to negative
(4) by putting
Mn, r)=
F^^^
(r < 0) .
With this extension, the r e a d i l y proved formula
k
L(n, r) =
k
2 ( ) L(n-k, r - i )
i=0 1
if valid for a l l k such that 0 — k < n.
r
from
1965
ON A CLASS OF NONLINEAR BINOMIAL SUMS
298
The author would like to thank Vincent C. H a r r i s for his helpful
suggestions.
REFERENCES
1.
David Dickinson,
On Sums Involving
A m e r i c a n M a t h e m a t i c a l Monthly,
2.
Mark Feinberg,
1964,
3.
Binomial
Coefficients,
Vol. 57, 1950, pp. 8 2 - 8 6 .
New Slants, The Fibonacci Q u a r t e r l y , Vol. 2,
pp. 223-227.
V. C. H a r r i s and Carolyn C. Styles, A G e n e r a l i z a t i o n of F i b onacci N u m b e r s ,
The Fibonacci Q u a r t e r l y ,
Vol. 2, 1964, pp.
277-289.
4.
Melvin H o c h s t e r , F i b o n a c c i - t y p e S e r i e s and P a s c a l ' s T r i a n g l e ,
5*
P a r t i c l e , VoL IV, 1962, pp. 14-28.
J o s e p h A. Raab, A G e n e r a l i z a t i o n of the Connection Between the
F i b o n a c c i Sequence and P a s c a l ' s T r i a n g l e , The F i b o n a c c i Q u a r t e r l y , Vol. 1, October, 1963, pp. 2 1 - 3 1 .
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The F i b o n a c c i Bibliographical R e s e a r c h Center d e s i r e s that any
r e a d e r finding a Fibonacci r e f e r e n c e , send a c a r d giving the r e f e r e n c e
and a brief d e s c r i p t i o n of the c o n t e n t s . P l e a s e forward all such inf o r m a t i o n to:
F i b o n a c c i Bibliographical R e s e a r c h Center,
Mathematics Department,
San J o s e State College,
San J o s e , California