This document summarizes properties of generalized Fibonacci numbers G,, defined by the recurrence relation G,=G,-r+G,-z. It presents three key points: 1) Matrices with columns containing consecutive G, terms have determinants of 1 or -1, allowing simple derivations of identities for G,. 2) Writing matrices in this way yields the identity G,+G,*r+G,*z=G,+a for any n. 3) Solving systems of equations using these matrices produces integral solutions and identities relating sums of consecutive G, terms to later terms in the sequence.

1. CLASSES OF IDENTITIES FOR THE GENERALIZFD FIBONACCI
NUMBERS G,, : Gr-r* G,-" FROM MATRICES WITH
CONSTANT VALUED DETBRMINANTS
Marj orie Bicknell-Johnson
665 Fairlane Avenue, Santa Clara, CA 9505 I
Colin Paul SPears
University 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, the sums of elements found on successive diagonals of Pascal's triangle written in left-
are
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 (1 1)
For the reader's convenience, the first values are listed below:
n01234 5 6 7 8 910
G"01112 3 4 6 9 13 79
n 1l 12 13 14 15 16 l7 18 19 20 2l
Gn 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 (l 2)
1392888277
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
19961 l2t

2. CLASSES OF IDENTITIES FOR THE GENERALZED FIBONACCI NUMBERS Gn = Gn r + G,-"
(t o o l
lo I o -, (1 3)
[o o I
I
+)
We note that, in any row, any four consecutive elements d, e,.f, and g are related by
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 "column
property" 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
G, + G, + G, + ... t G, = G,*3 - l, (1 s)
G, + Gu + Gn +'.' + Gtt, = G3k*, - l, (1.6)
G,+Go+G, +'..*G, *t= G3k*2, (1 7)
Gr+ G, + Gs + '.. * Gzt*z = Gzt,*2, (1 8)
which compare with
Fr+ Fr+ Fr+'..* Fr= Fn*2-1, (l e)
Fr+ Fr+ 4 +...* Fro*, = F2k*2, (1. lo)
Fr+ Fo+ Fu+..'* Frn = Fzk*r-1, (l.n)
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 of
rows, 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):
(c, G,*, G,*n )
4, =l Gr*t Gn*p*t G,*qt I
(2 t)
[Gr-, Gn+p+z Gr*q*z)
We can form matrix An*rby applying (l.l), replacing row I by (row I * row 3) in A" followed by
two row exchanges, so that
r22 IMAY
i

3. 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)
'r[i
Ai=lr 1 2 al+10 I 0 ll
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
+ BG n*, * TG,*z = G,+s.
1oG*,
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.
19961 t23
F

4. 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 with
G,*, 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)
1 I
4=lt 1 2 G,*, 1+lt o o G._rl-+lo o o
o G*_rl,
[r t G**z) [o I G,_,) [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,
F' 4'.n )
.1, =( F,*r*,
4*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,
* _(F, F,+z 4*, )
a. - Fn+r
F,u 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

5. cLAssES oF IDENTITIES FoR THE GENERALZED FIBoNACCI NUMBERS G, = G,-r+ G,-"
Gn+r2 = G, - 2Gn*4 * 5Gras, (2.r1)
where we compare (1.4) and (2.10).
For the Fibonacci numbers, solve
n =(F.., F,:: F,::)
bytakingn=1,
,i=(1 3 0 -l)
;)-(; | 3)'
so that
+q= -+3F,*". (2.r2)
Similarly,
Fr*e = Fr+4Fr4, (2.r3)
(2.14)
4*t= -4+71*o'
In the Fibonacci case, we can solve directly for Fn*ro from
,t+ (r, F,*, 4*ro )
4=
[4*' F*p*t Fn*zp*t )
by taking n= -1,
(F,F"';i;;i'r'r)- 0 (-l)o-')
=(; 7,' 4o-'l-
Fro ) [; |
'r' [; 1 Lo)'
since FrFro_r- FoqFzp = (-l)utF, and Fro = FoLo are known identities for the Fibonacci and
Lucas numbers. Thus,
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
(o r r Gr,-,) (o I I Gr.-, ) 1,o I I Gr,-'')
,41,=lot 2 Gr, l-lo o I Gr,-Gr.,_,1-lo o I Gr,-tl
[r I 3 G"") t',.,0
). , : ?: ,';,',,,
-+lo o I Gr*_, l-lo I o Gr,_r-Gr*_rl.
,' l";l- , )
[t ooGr,_r-Gr,_r) [oolGr,_,)
obtaining
Gn+2, = Gr,-rG, + (Gz.-r - Gz,-z)G r*z + Gr*-rG nno
(2 t6)
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6. cLAssES oF IDENTITIES FoR THE GENERALZED FIBoNAccl NUMBERS G, = G,-t + G,-"
In the Fibonacci case, taking n= -1,
I o
o _(F, 4*, 1+2, ) r o-' _(l | 4,-,')--fl I -4.-r)I
,* = Fn+z ,. =
l{u i,*r,*r)- (.0 Fr, J - (.o Fr,
we have
Fn+2, = - Frr_r4 + FzrFn+2 - (2.17)
Returning to (2.6) and (2.16), the same procedure leads to
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
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,
t
laGr+ BGo*r 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*, p+r G 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 = GzoGtp-r- GroGro-r.
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.
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7. 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,
(c, Gn*, G,*n )
B, =l Gr*z Gn*p+z Gn*q*z l,
[Gr.o Gn*p++ Gr*n*o )
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.,
G, + G, + G3 + ... + G* = G"(k+r) - l, (3 2)
which can be proved by induction. It is also true that
G+G,+q+"'*G,=Gn*"-I. (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 of
the 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, (3.s)
G^ + G"*. I G2"*.+'.. + G"k+^ = G"k+.+l, m = 1,2,..., c *1, (3 6)
for total of c related identities reminiscent of (1.6), (1.7), and (1.8).
a
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 I
by (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
lee6l 127
I

8. 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. Ifwewritethefirstrowof
4r," 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 write
identities 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,"=+l
whenever 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 G r+. - G rG.- + G r+rc.- + G r+2G r- -r +''' + Gn a
"*t " " "-1G, - "a2,
^a
L-L Fr+3= Fr+Fr*r*Fr*r,
c=3 Gr+4 = Gr+ Gn*r* Gr*2,
c=4 Gr+5= Gn+Gr*rlGn*,
c=5 Gr+6 = G, + Gn*r* Gra4,
c=c 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
**n
t28 Ivav