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A GENERALIZED PASCAL'S TRIANGLE
C. K, WONG*tatid T. W. MADDOCKS**
University of Illinois, Urbana, Illinois 61801
1. INTRODUCTION
In the study of a combinatorial minimization problem related to multimodule computer memory organizations [ 5 ] ,
a triangle of numbers is constructed, which enjoys many of the pleasant properties of Pascal's triangle [1,2]. These
numbers originate from counting a set of points in the /r-dimensional Euclidean space.
In this paper we only list some of the properties which are similar to those associated with Pascal's triangle. Other
properties will be the subject of further investigation.
2. f-SPHERES IN RECTILINEAR METRIC
Let
k
U(rk) = \(x1,x2,',xk)\xi
integers, / =
1,2, - A
The aim of this section is to obtain a formula for the cardinality \Ur
)
and ]|T) \x,-\ < r >.
| of the set Ur
.
Lemma 1. Let
integers, / =
(xi,X2,",xk)\xj
then
1,2,
h
>(k),
\S
•, k and ] P
\xs\ = j > ,
i = o,
_ J k-1
x
i=0
Proof.
Notethat the number of waysto place/ nondistinct objects into k distinct cells is f
k+
k~1
1
)• (See [3].)
Consequently, the number of ways to place/nondistinct objects into k distinct cells such that none of them is empty
is
( k~-\
'* ' n ^J ' '* vv6 group together all points (xl,x2,—^(f()
whicti have the same number of zero coordinates,
the result follows.
Theorem 1.
\U<rk>\ = h [ k i ) { k L i ) 2 k - i .
i=0
Proof.
It follows from
\u<k)\=i+±\sr>\
and ±(i)=
(-;).
*On leave f r o m IBM T.J. Watson Research Center, Y o r k t o w n Heights, N.Y. Supported in part by NSF GJ 31222.
* * N S F Academic Year Institute University of Illinois Fellow in Mathematics, 1972-73.
tCurrent address: I.B.M. T.J. Watson Research Center, Y o r k t o w n Heights, N.Y. 10598.
134
APR.1975
A GENERALIZED PASCAL'S TRIANGLE
The numbers \sj
| and \Ur\
135
have the following geometric interpretation:
It suffices to mention the case k = 2. Partition the Euclidean plane into unit squares. Fix any square as the origin,
which will be called the 0nth sphere. All squares which have at least one edge in common with the origin form the / st
sphere. All those with at least one edge in common with a square in the 1st sphere form the 2nd sphere, and so on.
3
3 2 3
3 2 1 2 3
3 2 10 1 2 3
3 2 1 2 3
3 2 3
3
Figure 1
The numbers in Fig. 1 indicate what spheres the squares are in. \Sj(2) i| is then the number of squares comprising
thefh sphere, i.e. its "surface area," and \Ur• | is the number of squares constituting the rth sphere and its interior,
i.e., its "volume."
The generalization to k> 2 is clear.
3. A GENERALIZED PASCAL'S TRIANGLE
For simplicity, let us write Mkj for |U ( r k) \. We then have the following observations:
Theorem 2.
(i)
Mkff = Mrfk
r-1
M
('•>
k+1,r = 2 ^
Mkj + Mkfr
1=0
(iii)
Proof.
Mk+hr+1
= Mk+1tr +
Mktr+i+Mkfr
(i) If k > r, then
^-E(n(-/)^ / = 2:(/--)f-/)^-s(;
1=0
1=0
r-j
?M = M ,kr
i=k-r
Similarly for the case k < r.
r-1
k
l=o
i=o
^E^-E(f)(*/i-/)^w
(ii)
r-1
/=/
1=0
1=0
( f c r)(/ + ;)^ , + ?[(?) + (/^)](*/r-/)^ W -;^/)(5i^
k+1
sCDU/-/)^"-"**^
i=0
(iii) It follows directly from (ii).
Theorem 3. For k = 0, 1, 2, - , let
136
A GENERALIZED PASCAL'S TRIANGLE
APR. 1975
(k-D/2
M
if k is odd,
k~2i,i,
if k is even.
H
Sk
Then
k-2i,h
k/2
M
X)
i=0
S
, , n < 0
k+3 =
for k = 0, 1,2, - .
In other words, they form a Tribonacci sequence.
$k+Sk+1+Sk+2
Proof. St follows from (iii) of Theorem 2.
We can now construct a triangle of the numbers | MkA,
| Mn-jj
k,r = 0, 1, 2, —. The nth row consists of the numbers
\ in the order of i = 0, 1,2, —,n from left to right. The left diagonals thus consist of numbers with fixed A;
and the right diagonals numbers with fixed k.
By (iii) of Theorem 2, each number in the n row is the sum of the three adjacent numbers in the (n - 1) and
(n - 2)nd rows. For example, the number 25 in the 5 t h row is the sum of its three adjacent numbers 5, 7,13 in the
3 rc! and 4 t h rows. Therefore, instead of using the formula in Theorem 1, we can fill in a row by adding appropriate
numbers in the two preceding rows. Finally, by Theorem 3, the sums of the more gently sloping diagonals form the
Tribonacci sequence, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, •••.
The first 10 rows of this generalized Pascal's triangle is displayed below.
r=0
Rows
Tribonacci
*1
0
Sequence
1
2
3
4
5
6
7
ACKNOWLEDGEMENT
The authors are grateful to Professor V.E. Hoggatt, Jr., for pointing out that in Professor Monte Boisen, Jr.'s article
[ 4 ] , the same triangle appeared but its generation was different.
REFERENCES
1.
2.
3.
4.
5.
M. Gardner, "The Multiple Charms of Pascal's Triangle," Scientific American, Dec. 1966, pp. 128-132.
D.E. Knuth, The Art of Computer Programming, Vol. 1, Addison Wesley, Reading, Mass., 1968.
C.L Liu, Introduction to Combinatorial Mathematics, McGraw-Hill, New York, 1968.
M. Boisen, Jr., "Overlays of Pascal's Triangle," The Fibonacci Quarterly, Vol. 7, No. 2 (Apr. 1969), pp. 131-139.
C.K. Wong and D. Coppersmith, " A Combinatorial Problem Related to Multimodule Memory Organization"
J. ACM, Vol. 21, No. 3 (July 1974), pp. 392-402.