applications of fibonacci numbers || the generalized fibonacci numbers {cn}, cn = cn-1 + cn-2 + k
TRANSCRIPT
Marjorie Bicknell-Johnson and Gerald E. Bergum
THE GENERALIZED FIBONACCI NUMBERS {Cn}, Cn = C n- 1 + Cn-2 + K
INTRODUCTION
There are many ways to generalize the well-known Fibonacci sequence {Fn },
Fn - Fn- 1 + Fn-2, Fl = I, F2 = 1. In a personal letter dated December 18, 1985, Frank
Harary asked one of the authors if they had ever encountered Cn - Cn- 1 + Cn- 2 +
I, which was used by Harary in connection with something he was counting
involving Boolean Algebras. In fact, in Harary's research it was noticed that the
value of one could be replaced by any integer k.
Furthermore, one of the authors noticed recently that the integers generated
by the sequence, with Co = C1 = I, are related to the number of nodes found in a
Fibonacci tree, [1], [2].
These observations culminated into the output of this paper and some
questions with unknown answers.
1. SPECIAL SEQUENCES
If we generalize the Fibonacci sequence {Fn} = {I, I, 2, 3, 5, 8, ••• } by adding
one at each step of the recursion, we form a new sequence {Cn} = {I, I, 3, 50 9, IS,
25, 41, .•. } where
cn+1 = Cn + C n -l + I, C 1 = I, C2 = 1.
It is not difficult to show that
cn=2Fn-1,
193
A. N. Philippou et al. (cds.), Applications of Fibonacci Numbers,193-205. © 1988 by Kluwer Academic Publishers.
(Ll)
(1.2)
194 M. JOHNSON AND G. E. BERGUM
and to extend the subscripts so that n is any integer.
However, more fruitful special sequences can be formed by changing the
beginning values in (1.1). If we write {c!} - {t, 2, 4, 7, 12, 20, 33, 54, 88, ••• } where
c! - 1 and c; = 2 in the recursion of (1.1), we find that
(1.3)
Changing the initial values to zero, results in {c~} = {O, 0, I, 2, 4, 7, 12, 20, •.• }
which has the same terms occurring as in {c!} but
c~ - Fn - 1. (1.4)
Actually, the sequences {Cn}, {c!}, and {c~} are all special cases of the more
general sequence (Cn(a,b)}, where
The first few terms of (Cn(a,b)} are
a, b, a + b + I, a + 2b + 2, 2a + 3b + 4, 3a + 5b + 7, Sa + 8b + 12, •..
Cn(a,b) = aFn-2 + bFn-1 + Fn - I, D~ I, (1.6)
and by standard techniques the generating function g(a,b;x) for (Cn(a,b)} is
g(a,b;x) _ !: Cn(a,b)xn- 1 = a + (b - 2a)x + (a - b + 1)x2 (1.7) n = 1 1 - 2x + x3
Since a and b are arbitrary. one could investigate many possibilities. We Dote three
cases that we found to be most interesting.
When a = b = 1. (Cn (1.l)} is generated by
gO l·x) = 1 - x + x2
• • 1 - 2x + x3 '
THE GENERALIZED FIBONACCI NUMBERS .••
and, as in (1.2),
Cn (1,1) = 2F .. - 1.
If a = 1 and b - 2, then
g(1,2;x) = 1 1 - 2x + x3
and, as in (1.3),
with
Also, note that for a = b = 0, C .. (O,O) = c~ from (1.4), and
g(O O.x) = x2
, , 1 - 2x + x3 '
Cn(O,O) = F n - 1.
In passing, we note that, since
limit CnH(a,b) n --.00 Cn(atb)
limit F n+1t = [1 + ..J5)1t n--+oo Fn 2 '
1 + .JS -2-
195
(1.8)
(1.9)
(1.10)
(1.11)
One can easily extend {Cn(a,b)} to negative subscripts, and show that (1.8)
holds for all integers n. Then, since Co(a,b) = b - a-I and C_1(a,b) = 2a - b, we can
rewrite the generating function for {Cn(a,b)} from (1.9) as
( b. ) = ~ C ( b) n-I = a - C_1(a,b)x - Co(a,b)x2
g a, ,x ,£...., n a, x 1 _ 2x + x3 n = 1
Furthermore, we note that (Cn(1,2)} and (Cn(O,O)} are important special cases,
since (1.6) can be rewritten as
Cn(a,b) = aF n-2 + bF n-I + Cn-2(1,2) (1.12)
196 M. JOHNSON AND G. E. BERGUM
or
C,,(a,b) = aF ,,-z + bF ,,-1 + C,,(O,O).
In fact, (1.12) and (1.13) lead to multitudinous special cases:
C"O,2) - F"+1 + C .. -2(1,2)
C,,(O,O) = C,,-z(1,2)
C,,(O,b) = bF ,,-1 + C,,(O,O)
C,,(a,O) - aF .. -2 + C .. (O,O)
C .. (a,a) = aF.. + C .. (O,O)
C .. (a,2a) = aF .. +1 + C .. CO,O)
C .. (-a,a) = aF .. _a + C .. (O,O)
C .. (F .. ,F"+1) = F .. + .. -1 + C .. (O,O)
Since the Lucas sequence {L .. } is given by
we have the following additional special cases:
C .. (2,1) = L .. - 1 + C .. (O.O)
C .. (3,1) = 2L"-1 - 1
C .. O,3) = l .. + C .. (O.O) = 2F"+1 - 1
C,,(2.3) = L"+1 - 1
Cn (3,4) = Ln+l + C .. CO,O)
C .. (2a,a) = aL .. _1 + C .. (O,O)
C,,(a,3a) = aL .. + Cn(O,O)
C .. (-a,2a) = aLn-2 + Cn(O,O)
Cn(l,., L"'+1) = L .. + .. -1 + C,,(O,O)
Returning to (1.6), we can easily derive
C,,(a+l,b+l) = F .. + Cn(a.b)
0.13)
(1.14)
(1.15)
(1.16)
(1.17)
(1.18)
0.19)
0.20)
0.21)
(1.22)
(1.23)
(1.24)
(1.25)
0.26)
0.27)
0.28)
0.29)
(1.30)
(1.31)
THE GENERALIZED FIBONACCI NUMBERS .•.
and
C,,(a+m,b+m) = mF" + C,,(a,b).
Using (1.14), we can change (1.12) to
so that
leading to
C,,(a,b) = (a - OF ,,-2 + (b - 2)F ,,-I + C"U,2)
C"U,b) = (b - 2)F"_1 + C"U,2)
C,,(a,2) = (a - OF ,,-2 + C"U,2)
C,,(a,2a) = (a - OF 'HI + C"U,2)
C,,(3,O = F"-4 + C"U,2)
C,,(3,4) = 2F" + C"U,2)
C,,(a,a+1) ~ (a - OF" + C"U,2)
197
U.32)
(1.33)
U.34)
(1.35)
(1.36)
U.37)
U.38)
(1.39)
No attempt has been made to be exhaustive, but it is time to turn to some special
summations.
Since
C"O,2) is the sum of the first n Fibonacci numbers. By using (1.6) and U.40),
Thus,
n n R R L CIt.(a,b) = a L FIt.-2 + b L FIt.-1 + L Fit. - n It. = I It.-I It.-I It.-I
= a f Fit. + b f + ~ Fit. - n It. = -I It. = I It. = I
= a(F_1 + Fo + F" - 0 + b(F"+l - 1) + (F"+2 - 0 - n
= (aF" + bF,,+1 + F"+2 - 1) - b - n
= Cn +2(a,b) - b - n.
U.40)
198 M. JOHNSON AND G. E. BERGUM
~ C,.(a,b) - C,,+:zCa,b) - (b + n) (1.41)
,. - I
with special cases
Since
~ C,.(1,2) - C .. +:zCl,2) - (n + 2) - F"+4 - (n + 3) II _ ,
~ c,.(1,1) - C .. +:zCl,1) - (n + 1) - 2F,,+2 - (n + 2) ,. - I
~ C,.(D,D) - c..+:zCD,D) - n - F"+2 - (n + 1) ,. - I
~ kF,.+p - (n + 1)F"+2+p - F"+4+P + F3+p, ,. - I
(1.42)
(1.43)
(1.44)
(1.4S)
we can use (1.6) to derive
so that
- a[(n + 1)F .. - F"+2 + Fil + b[(n+1)F"+1 - F"+3 + F2l
+[(n + 1)F"+2 - F"+4 + F3l - n(n + 1)12
- (n + lXaF .. + bF"+l + F"+2)
- (aF .. +2 + bF"+3 + F,,+4) + (aFI + bF2 + F3) - n(n + 1)12
- (n + lXC,,+:zCa,b) + 1] - [C"+4(a,b) + 1]
+ [C3(a,b) + 1] - n(n+1)12
- (n + 1)C"+2(a,b) - C"+4(a,b) + C3(a,b) - (n - 2Xn + 1)12
~ kc,.(a,b) - (n + 1)C"+2(a,b) - C"+4(a,b) + C3(a,b) (1.46) ,. - I
(n - 2Xn + 1) 2
THE GENERALIZED FIBONACCI NUMBERS • • • 199
Looking to
(1.47)
with patient algebra, one can derive
C~(a,b) - Cn - 1(a,b)C,,+l(a,b) = (a2 - b2 + 3a + ab - b + 1)(_1)"+1
+ Cn - 3(a,b) + 1 (1.48)
from 0.6).
Suppose that we put {Cn(a,b)} into a 3x3 determinant as below, where we let
Cn = C,,(a,b). Using (1.5), we subtract the sum of the first and second columns from
the third column, followed by the same sequence for the rows. Then, add the third
column to the first and second columns.
On =
Cn Cn+1
Cn+l Cn+2
Cn+2 Cn+3
C"+2 Cn C"+l 1 C" C"+1
Cn+3 Cn+l Cn+2 1 = Cn+t Cn+2
Cn+1 Cn+2 Cn+3 1 1 1
Cn + 1 Cn+t + 1 1
Cn+l + 1 Cn+2 + 1 1
0 0 -I
= (-I)[(Cn + l)(Cn+2 + 1) - (Cn+t + 1)2]
- (C!+t + 2Cn+t + 1) - (C"Cn+2 + Cn + Cn +2 + 1)
= C!H - Cn Cn+2 + (C"+1 - Cn) - (Cn+2 - CnH)·
Returning to (l.5), notice that
and continue to derive On.
1
1
-1
(1.49)
200 M. JOHNSON AND G. E. BERGUM
D .. - (C!+l - C"C .. +2) + (C .. _I + 1) - (C .. - 1)
- (C!+I - C"C,,+2) - C .. -2 - 1
[(a2 _ b2 + 3a + ab - b + lX-U" + C,,-z + 11 - C"-2 - 1
by (1.48). Hence, we have
D" - (a2 - b2 + 3a + ab - b + lX-U". (1.50)
Note that, if we use {C .. (1,U} to form D", then D" - 4<-U", while D .. - (-U" for
{C .. (O,O)} or for {C.(1,2)}.
2. THE GENERALIZED SEQUENCE C.(a,b,k)
When we let C,,(a,b,k) be defined as
C,,(a,b,k) - C"_I(a,b,k) + C"_2(a,b,k) + k, CI = a, C2 = b, (2.1)
the first few terms
a, b, a + b + k, a + 2b + 2k. 2a + 3b + 4k. 3a + Sb + 7k.
immediately suggest
Sa + 8b + 12k. 8a + 13b + :lOk, ••• ,
C,,(a,b,k) - aF ,,-2 + bF .. _I + kC .. (O,O),
C,,(a,b,k) - aF .. -2 + bF ,,_I + kF.. - k,
where the subscripts can extend to any integer. Of course, C .. (a,b,O - C .. (a,b).
Applying (1.40) to (2.3), one can optain
n
(2.2)
(2.3)
L Ct(a,b,k) - aF" + bF"+l + kF"+2 - b - (n + Uk (2.4)
t - I _ C"+2(a,b,k) _ b - kn.
THE GENERALIZED FIBONACCI NUMBERS •••
Since !i!!~ F;:1t = (1 +2.fsl" ' elementary limit theorems lead to
Using
limit CnH(a,b,k) 1 + .fs n ->(x> Cn(a,b,k) = --2-
C .. (O,O) = CnU,2) - F"+1
201
(2.5)
from (1.14) and (1.15) in (2.2), one can convert C .. (a,b,k) into a form in which C .. U,2)
appears, writing
C .. (a,b,k) = (a - k)F .. -2 + (b - 2k)F .. _1 + kC .. (1,2). (2.6)
Next, we use (2.2) and (2.6) to write some particular sequences {C .. (a,b,k)} that have
interesting general terms.
C,.(a,a,k) = aF.. + kC .. (O,O)
Cn(a,O,k) = aF n-2 + kC .. (O,O)
Cn(O,b,k) = bF .. -I + kC .. (O,O)
C .. (a,2a,k) = aF n+1 + kC .. (O,O)
Cn(F .. ,F",H,k) = F.+n-1 + kCn(O,O)
C .. (L .. ,L .. H,k) = L.+ .. -1 + kCn(O,O)
Cn(a+m,b+m,k) = mF,. + C .. (a.b.k)
Cn(k.b,k) = (b - 2k)Fn-1 + kCnU,2)
Cn(a,2k,k) = (a - k)F n-2 + kC .. (1,2)
Cn(b-k,b,k) = (b - 2k)F .. + kCnO,2)
Cn(a,2a,k) = (a - k)F "+1 + kCnU,2)
Cn(a,a-k.k) = aFn + kCn_2 (O,O)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
If {Cn(a,b,k)} is put into a 3x3 determinant Dn, where we let C .. = C .. (a,b,k), we
can establish that
202 M. JOHNSON AND G. E. BERGUM
= (_l)n k(a2 _ b2 + k2 + 3ak + ab - bk). (2.19)
Since Cn- 1 + Cn - Cft+! - k,
Cn Cn+! C"+2 Cn Cn+! k
Dn = Cn+! Cn+2 Cn+3 Cn+! Cn+2 k
Cn+2 Cn+3 Cn+4 C"+2 Cn+3 k
Since Cn - C .. +! - -C .. -1 - k,
C .. - 1 C .. Cn+! -k Cn Cn+!
Dn - 1 - C" Cn+! Cn+2 -k Cn+! Cn +2
C"+ l C"+2 Cn+3 -k Cn+2 Cn+8
and Dn - 1 = (-l)Dn. So, we only have to evaluate one of these determinants Dn.
-a+b-k a b -a+b-k a k
Do = a b a+b+k a b k
b a+b+k a+2b+2k k k -k
-a+b b+k 0
a+k b+k 0
k k -k
= k(a2 _ b2 + k2 + 3ak + ab - bk).
Thus, since Dn - 1 = (-l)Dn, we have (2.19).
Since we also could write
THE GENERALIZED FIBONACCI NUMBERS .•.
Cn- 1 Cn k Cn- 1 + k Cn + k
-: I Dn- 1 = Cn Cn+! k Cn + k Cn- 1 + k
k k -k k k
= k[(Cn + k)2 - (C"-l + kXCn+1 + k)],
we can derive another identity with a little patience. Observe that
since
(Cn + k)2 - (Cn - 1 + k)(Cn+! + k)
= (C~ - Cn- 1Cn+1) - k(Cn +1 + Cn- 1 - len)
= (C~ - C"-lCn+1) - k(C,,-3 + k)
Thus, rearranging (2.21) and using (2.19) and (2.20),
203
(2.20)
(2.21)
(2.22)
C~ - C,,+!C"-1 = (Cn + kf - (Cn+! + k)(Cn-1 + k) + k(Cn-3 + k)
- (D"_I)/k + k(Cn - 3 + k)
establishing
C~ - C"+1C"-l = (_On-l(a2 - b2 + k2 + Jak + ab - bk)
+ kCn-3 + k2
where Cn = Cn(a,b,k). Compare with Equation 0.48).
(2.23)
Finally, we make a sum where we lean on Equation (l.45) several times,
writing
= a[(n + OFn - Fn+2 - 1] + b[(n + OF,,+! - Fn+3 + 1]
+k[(n + OF ,,+2 - F n+1 + 2] - kn(n + 1)12
= (n + l)(aFn + bF"+1 + kFn+2 - k) + (n + Ok
- (aFn+2 + bFn+3 + kFn+1 - k) - k
M. JOHNSON AND G. E. BERGUM
+ (a + b + k) + k - kn(n + 012
= (n + 1)C"+2 - C"+4 + C3 + (n+Ok - kn(n + 012
finally yielding
'i: iCt(a,b,k) - (n + l)C"+2(a,b,k) - C"+4(a,b,k) + C3(a,b,k) (2.24) t - 1
ken - 2)(n + 0 2
which has Equation (1.46) as a special case.
Similarly to (1.40, one could write
" :L: Ct(a,b,k) = C,,+2(a,b,k) - b - kn. t = 1
3. FURTHER POSSIBLE INVESTIGATIONS
(2.25)
It should now be obvious that many of the results known about the Fibonacci
sequence can be related to the sequence given in (2.0. To continue this
development would be a trivial pursuit. However, there are several challenging
questions which the authors would like to propose to the reader:
(0 C;:an one develop divisibility properties similar to those known about the
Fibonacci sequence?
(2) What is the value of the largest perfect square in anyone of the
generalized sequences?
(3) Do triangular numbers appear more frequently in the generalized
sequences than in the Fibonacci sequence?
(4) Are the sequences complete from the integer representation point of
view?
(5) How are the roots of the equation x" = X"-l + X"-2 + 1 related to
numbers of the sequence in (1.0?
THE GENERALIZED FIBONACCI NUMBERS •••
REFERENCES
[1J Horibe, Y. "An Entropy View of Fibonacci Trees." The Fibonacci Quarterly.
Vol. 20. No.2 (1982): pp 168-178.
[2J Horibe, Y. "Notes on Fibonacci Trees and Their Optimality." The Fibonacci
Quarterly. Vol. 21. No.2 (1983): pp 118-128.
205