CLASSES OF IDENTITIES FOR THE GENERALIZFD FIBONACCINUMBERS G,, : Gr-r* G,-" FROM MATRICES WITH

CONSTANT VALUED DETBRMINANTS

Marj orie Bicknell-Johnson665 Fairlane Avenue, Santa Clara, CA 9505 I

Colin Paul SPearsUniversity of Southem California, Cancer Research Laboratory, Room 206

1303 North Mission Road, Los Angeles, CA 90033

(Submitted June 1994)

The generalized Fibonacci numbers {G,},Gr=G,-r*G,-",n} c,Go=o,Gt- Gz="'=G"-l

=I, are the sums of elements found on successive diagonals of Pascal's triangle written in left-

justified form, by beginning in the left-most column and moving up (c- 1) and right 1 throughout

the array [1]. Of course, G,=Fn,the nth Fibonacci number, when c=2. Also, G, =u(n-l;c- 1,1), where u(n; p,q) are the generalized Fibonacci numbers of Harris and Styles [2]. In this

paper, elementary matrix operations make simple derivations of entire classes of identities for such

generalizedFibonacci numbers, and for the Fibonacci numbers themselves.

I. INTRODUCTION

Begin with the sequence {G,}, such that

Gr=Gn-t*Gn-3, n>3, Go = 0, Gr =Gz=1

For the reader's convenience, the first values are listed below:

(1 1)

5

3

7

6

91013 79

n 1l 12 13 14 15 16 l7 18 19 20 2lGn 28 41 60 88 129 189 277 406 595 872 1278

These numbers can be generated by a simple scheme from an array which has 0, l, 1 in the first

column, and which is formed by taking each successive element as the sum of the element above

and the element to the left in the array, except that in the case of an element in the first row we

use the last term in the preceding column and the element to the left:

o 1 4 13 [41]J rze| 2 6 [9]+ 60 189

1392888277(l 2)

If we choose a 3 x 3 array from any three consecutive columns, the determinant is l. If any 3 x 4

array is chosen with 4 consecutive columns, and row reduced by elementary matrix methods, the

solution is

8

9

6

4n01234

G"01112

19961 l2t

CLASSES OF IDENTITIES FOR THE GENERALZED FIBONACCI NUMBERS Gn = Gn r + G,-"

(t o o l\lo I o -, I

[o o I +)

We note that, in any row, any four consecutive elements d, e,.f, and g are related by

(1 3)

(1 s)

(1.6)

(1 7)

(1 8)

(l e)

(1. lo)

(l.n)

d-3e+4f = g. (1 4)

Each element in the third row is one more than the sum of the 3fr elements in the k preceding

columns; i.e.,9=(3+2+1+l+l+0)+1. Each element in the second row satisfies a "columnproperty" as the sum of the tlree elements in the preceding column; i.e., 60=28+19+13 or,

alternately, a "row property" as each element in the second row is one more than the sum of the

elementaboveandallotherelementsinthefirstrow;i.e.,69=(41+13+4+l+0)+1. Eachele-ment in the third row is the sum of the element above and all other elements to the left in the

second row; i.e., 28 = 19 + 6+2+ l. It can be proved by induction that

which compare with

G, + G, + G, + ... t G, = G,*3 - l,

G, + Gu + Gn +'.' + Gtt, = G3k*, - l,

G,+Go+G, +'..*G, *t= G3k*2,

Gr+ G, + Gs + '.. * Gzt*z = Gzt,*2,

Fr+ Fr+ Fr+'..* Fr= Fn*2-1,

Fr+ Fr+ 4 +...* Fro*, = F2k*2,

Fr+ Fo+ Fu+..'* Frn = Fzk*r-1,

IMAY

for Fibonacci numbers

The reader should note that forming a two-rowed array analogous to (1.2) by taking 0, 1 in

the first column yields Fibonacci numbers, while taking 0, l, l, ..., with an infinite number ofrows, forms Pascal's triangle in rectangular form, bordered on the top by a row of zeros. We also

note that all of these sequences could be generated by taking the first column as all I 's or as L, 2,

3, ..., or as the appropriate number of consecutive terms in the sequence. They all satisfy "row

properties" and "column properties." The determinant and matrix properties observed in (1.2)

and (1.3) lead to entire classes ofidentities in the next section.

2. IDENTITMS FOR THE FIBONACCI NUMBERS AND FOR THE CASE C:3

Write a 3 x 3 matrix ,4" = (o) by writing three consecutive terms of {G,} in each column and

taking arr= G,,where c = 3 as in (1.1):

We can form matrix

two row exchanges,

r22

(2 t)

I * row 3) in A" followed by

(c, G,*, G,*n )4, =l Gr*t Gn*p*t G,*qt

I

[Gr-, Gn+p+z Gr*q*z)

An*rby applying (l.l), replacing row I by (row

so that

i

CLASSES OF IDENTITIES FORTHE GENERALIZED FIBONACCI NIJMBERS G, = G, ,*G,.

det\=detAn*r. (22)

Letn=7,p=l,e=2 in(2.1) and find det.4t = -1. Thus, det\= -1 for

(G, G,+t G,*r)4 =l G,*, Gn+z G*, | (2 3)

[G,-, Gn+t G*o)

As another special case of (2. 1), use 9 consecutive elements of {G,} to write

(G, Gn*t G*u)4, =l G,*, G,+q Gn*t l, Q 4)

[G,., G,+s G*r)

which has det \ = 1.

These simple observations allow us to write many identities for {G,} effortlessly. We

illustrate our procedure with an example. Suppose we want an identity of the form

aG, + BGn*r* TGr+z = G,+4.

We write an augmented matrix .{,, where each column contains three consecutive elements of

{G"} and where the first row contains G* G,*1, G,*r, and Gn*o'.

(c, G,+r G,*z G,*o)

4 =l G,u G,*z G,*t G,*s I

[G,., Gr*t Gn+q G*o)

Then take a convenient value for n, say n=1, and use elementary row operations on the aug-

mented matrix Ai,(t l I 3) (t o o l)

Ai=lr 1 2 al+10 I 0 ll'r[i t16) [ooii)to obtain a generalization of the "column property" of the introduction,

Gr+Gr*r*Gr*z= Gr+4, Q 5\

which holds for any n.

While we are using matrix methods to solve the system

laG, + BG,*t t lGn+z = C,+4,I

1oG*, + BG n*, * TG,*z = G,+s.

lac n*2 * N r+t * /Gn*q = Gn+6,

notice that each determinant that would be used in a solution by Cramer's rule is of the form

det\= det Ar*, from (2.1) and (2.2), and, moreover, the determinant of coefficients equals -l so

that there will be integral solutions. Alternately, by (l . 1), notice that (a, f , y\ will be a solution

of aGn*r*FG,*q*TGn+s=G,+7 whenever (a,F,y) is a solution of the system above for any

n > 0 so that we solve all such equations whenever we have a solution for any three consecutive

values of n.

F

19961 t23

cLASSES oF IDENTITIES FoR THE GENERALZED FIBoNAccI NUMBERS G, = G,,, + G,,"

We could make one identity at a time by augmenting An with a fourth column beginning withG,*, for any pleasing value for w, except that w < 0 would force extension of {G,} to negative

subscripts. However, it is not difficult to solve

(G, Gn*r G,*z G,** )4 =l G,*, G,+z Gn*t Gn*,*r I

[Gr., C,+z 8n++ Gr***z )

by taking n = 0 and elementary row reduction, since G,+z - G.+t = G._rby (1. 1), and

(o r r G, ) (o I I G. ) (r o o c._r)4=lt 1 1 G,*, 1+lt o o G._rl-+lo I o G*_rl,

[r t 2 G**z) [o o I G,_,) [o o I G,_,)

so that

Gn** = G*_zGn+G.4Gn+r+G._rGr*r. (2 6)

For the Fibonacci numbers, we can use the malrix A,

.1, =( F' 4'.n )

\4*r F,*r*, )'for which det\=(-I)detA,*r. Of course, when q=1, det 4,=?l)'*t where, also, det\=4F,*r- F]*r, giving the well-known

(-1)'*t = FrFr*z- Fr'.. (2.7)

Solve the augmented matrix .{ as before,

a. _(F, Fn+r 4*, )* - \F,u F,+z 4*,*t)'bytaking n=-1,

1r,=(l ? ?,}to obtain

F,_r1+F,1*r= 4*.. (2 8)

Identities of the type aGn + BGn*, t TGn+c = Gn+6 can be obtained as before by row reduction

of(c, Gn+z G,+q e*u)

4 =l G,*, Gn*t Gn*s G,*, l.

[G,., G,+q Gr+a Gr*r )

If we take n=0, det 4=1, and we find a =1, p =2,y =1, so that

Gr+6 = G, +2Gr*, * Gr*c. Q 9)

In a similar manner, we can derive

Gr+e = Gn-3Gr$+4G,+6, Q.l})

t24 Iuev

l

cLAssES oF IDENTITIES FoR THE GENERALZED FIBoNACCI NUMBERS G, = G,-r+ G,-"

Gn+r2 = G, - 2Gn*4 * 5Gras,

where we compare (1.4) and (2.10).

For the Fibonacci numbers, solve

(2.r1)

(2.r2)

(2.r3)

(2.14)

bytakingn=1,

so that

n =(F.., F,:: F,::)

,i=(1 3 ;)-(;

,t+ (r, F,*, 4*ro )4 = [4*' F*p*t Fn*zp*t )

0 -l)| 3)'

\+q= -\+3F,*".Similarly,

Fr*e = Fr+4Fr4,

4*t= -4+71*o'In the Fibonacci case, we can solve directly for Fn*ro from

by taking n= -1,

since FrFro_r- FoqFzp =

Lucas numbers. Thus,

and Fro = FoLo are known identities

F,+z p = eI)P-' F, + LoF,* n.(2 ls)

Returning to (2.9), we can derive identities of the form a(Jn+fG,*z*TGn+q=G,*r. from

(2.1) with p = 2, Q = 4, taVingthe augmented matrix .'{, with first row containing G,, Gn+z, G,+4,

G,*2,. It is computationally advantageous to take n= -1', notice that we can define G-r = 0. We

make use of Gr. - Gr,-t = Gz,-3 from ( I . 1) to solve

obtaining

(2 t6)

l2-i

'r' =(; 7,' [; |

(F,F"';i;;i'r'r)- [;

4o-'l-Fro )

(-l)utF,

0 (-l)o-')1 Lo)'

for the Fibonacci and

(o r r Gr,-,) (o I I Gr.-, ) 1,o I I Gr,-''),41,=lo t 2 Gr, l-lo o I Gr,-Gr.,_,1-lo o I Gr,-tl

[r I 3 G"") ). , : ?: ,';,',,,

t',.,0 ,' l";l- , )-+lo o I Gr*_, l-lo I o Gr,_r-Gr*_rl.

[t ooGr,_r-Gr,_r) [oolGr,_,)

re96l

Gn+2, = Gr,-rG, + (Gz.-r - Gz,-z)G r*z + Gr*-rG nno

Gn+3,=Gr.-uGr+(Gt.-4G3r4)Gr$+G3,4Gn+6. (2.18)

The Fibonacci case, derived by taking n = -1,

/* ( F" Fr+t Fr+3, )* = [4*, F,*+ F,+t,+r)'

gives us

cLAssES oF IDENTITIES FoR THE GENERALZED FIBoNAccl NUMBERS G, = G,-t + G,-"

In the Fibonacci case, taking n= -1,

,* _(F, Fn+z 1+2, ) r ,. _(l I 4,-,')--fl o -4.-r)o = l{u 4*, i,*r,*r)- o-' = (.0 | Fr, J - (.o I Fr, Iwe have

Fn+2, = - Frr_r4 + FzrFn+2 -

Returning to (2.6) and (2.16), the same procedure leads to

(2.17)

4+t* = 1Fr,-, l2+ \*rFr, 12, (2.1e)

where F3^12 happens to be an integer for any m. Note that det\=(-l)"*t2, and hence,

detAn++1 . We cannot make a pleasing identity of the form aG,tFG,*q*TGn+s=Gn*o* for

arbitrary w because detAr+ +1, leading to nonintegral solutions. However, we can find an iden-

tity for {G,} analogous to (2.l5). We solve

faG_, + BG r-t t lGzp_t = Gtp-t,I

laGo+ No+ycro- Gtp,

laGr+ BGo*r t lGzp+r = Gtp*r.

for (u,F,y) by Cramer's rule. Note that the determinant of coefiicients D is given by D=G2oGot- GoGrur. Then a = A I D, where I is the determinant

lGro_, Gp_r Gro_,

A =lGr, Gp Gro

lGro*, G p+r Gzp+r

After making two column exchanges in A, we see from (2.I) and (2.2) that A : D, so a = I . Then

F = B / D, where B is the determinant

lo Gro-, Gro-,B = lo Gro Gro

lt Gtp+r Gzp*r

Similarly, y =C lD, where C=G3Got-GoG3o-r. Thus,

Gn+3p = G, + Gn*o(Grp-tGzp - GroGro-r) I D + G,*ro(GroGot- Gzp-S) I D,

where D= (G2rGo-r- GrGro-r). The coefficients of Gn+p and Gn*20 are integers for p - 1,2,...,9 ,

and it is conjectured that they are always integers.

= GzoGtp-r- GroGro-r.

t26 IMAY

CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G, = G, , + G,_"

As an observation before going to the general case, notice that identities such as (2.9), (2. l0),

and (Z.ll) generate more matrices with constant valued determinants. For example, (2.9) leads to

matrix fl,

where detB, = detBr*r.

3. THE GENERAL CASE: Gn = Gn-r* Gn-.

The general case for {G,} is defined by

Gr= Gn-t+Gr-", n) c,where Go = 0, Gt=Gz= "' = G"-t=l' (3 1)

To write the elements of {G,} simply, use an array of c rows with the first column containing 0

followed by ("- 1) l's, noting that 1, 2,3, ..., c will appear in the second column, analogous to the

array of (1.2). Take each term to be the sum of the term above and the term to the left, where we

drop below for elements in the first row as before. Atty c x c array formed from any c consecu-

tive columns will have a determinant value of +1. Each element in the cft row is one more than

the sum of the ck elements in the ft preceding columns, i.e.,

(c, Gn*, G,*n )B, =l Gr*z Gn*p+z Gn*q*z l,

[Gr.o Gn*p++ Gr*n*o )

G, + G, + G3 + ... + G* = G"(k+r) - l,

which can be proved by induction. It is also true that

G+G,+q+"'*G,=Gn*"-I.

(3 2)

(3 3)

Each array satisfies the "column property" of (2.5) in that each element in the (" - l)" row is

the sum of the c elements in the preceding column and, more generally. for any n,

G,+"-z = Gn-" * Gn-("-r\ + ". + Gn-z + G n-t (c terms) (3 4)

Each array has "row properties" such that each element in the lft row, 31i1c, is the sum ofthe element above and all other elements to the left in the (l - l)s row, while each element in the

second row is one more than the sum of the elements above and to the left in the first row, or

Go + G" + Gr" + Gr" + '.. + Gck = G"k*r- 1,

G^ + G"*. I G2"*.+'.. + G"k+^ = G"k+.+l, m = 1,2,..., c *1,

for a total of c related identities reminiscent of (1.6), (1.7), and (1.8).

The matrix properties of Section 2 also extend to the general case. Form the c x c matrix

An,"=(ar), where each column contains c consecutive elements of {G,} and arr=Gr. Then, as

in the case c = 3,

det Ar," = (-l)"-' det Aoq,", (3.7)

since each column satisfies Gr*" = Gn*"-r14. W" can form An*1." from An," by replacing row Iby (row I + row c) followed by (c- l) row exchanges.

When we take the special case in which the first row of A,." contains c consecutive elements

of {G"} , then Ar." = +1. The easiest way to justi$ this result is to observe that (3.1) cam be used

(3.s)

(3 6)

I

I

lee6l 127

CLASSES oF IDENTITIES FoR THE GENERALZED FIBoNAccI NUMBERS G, = G, | + G*"

to extend {G,} tonegative subscripts. In fact, inthe sequence {G,} extended by recursion (3.1),

Gr=landG,isfollowedby(c-l) l'sandprecededby(c-l)0's. Ifwewritethefirstrowof4r," as G*Gn-1,Gn-2,...,Gn-(c-t1, then, for n=l,the first row is 1, 0, 0, ..' 0. If each column

contains c consecutive increasing terms of {G,}, then G, appears on the main diagonal in every

row. Thus, .4,," has I's everywhere on the main diagonal with 0's everywhere above, so that

det A1,"=I. That det 4,,"=!l is significant, however, because it indicates that we can writeidentities following the same procedures as for c = 3, expecting integral results when solving

systems as before. Note that detAn,"=+l if the first row contains c consecutive elements of

{G,}, but order dt-res not matter. Also, we have the interesting special case that detAn,"=+lwhenever Ar," contains c2 consecutive terms of {Gr}, taken in either increasing or decreasing

order, c > 2. Det 4r," = 0 only if two elements in row I are equal, since any c consecutive germs

of G, are relatively prime [2].Again, solving an augmented matrix ,\." will make identities of the form

Gr*. = aoG, + dtGr+t + a rGn*2+ ... + d "-rGn+"-I

for different fixed values of c, or other classes of identities of your choosing. As examples, we

have:c =2 Fr+* = FrFr-t+ Fr+rF*c =3 Gn+w = GrGr-z+Gr+tcr4!Gra2Gr-1,c = 4 Gn+w = GnG.-t+Gr+tG.-4+Gr+2Gr-5*Gra3Gr-2,c = 5 Gn*. = GnG.-q+Gn+tcr-s+Gr+zcr-6+Gr$G*-7 *Gra4G*4,

c=c

^aL-L

c=3c=4c=5

c=c

G r+. - G rG.- "*t + G r+rc.-

" + G r+2G r-

" -r +''' + G

n a "-1G, - "a2,

Fr+3= Fr+Fr*r*Fr*r,Gr+4 = Gr+ Gn*r* Gr*2,

Gr+5= Gn+Gr*rlGn*,Gr+6 = G, + Gn*r* Gra4,

Gr+"+r = Gn + Gr*t r Gn*"-r.

So many identities, so little time!

REFERENCES

1. Marjorie Bicknell. "A Primer for the Fibonacci Numbers: Part VIII: Sequences of Sums

from Pascal's Triangle. " Ihe Fibonacci Quarterly 9.1 (197 l)'.1 4-81 .

2. V. C. Harris & Carolyn C. Styles. "A Generalization of Fibonacci Numbers." The Fibonacci

Quarterly 2.4 (19 64):27 7 -89 .

AMS Classification Numbers: l 1865, I1839, llc20

t28

**n

Ivav