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Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers

Yüksel Soykan

Department of Mathematics,

Art and Science Faculty,

Zonguldak Bülent Ecevit University,

67100, Zonguldak, Turkey

e-mail: [email protected]

Abstract. In this paper, closed forms of the summation formulas for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers.

2010 Mathematics Subject Classi…cation. 11B37, 11B39, 11B83.

Keywords. Fibonacci numbers, Gaussian Fibonacci numbers, Lucas numbers, Pell numbers, Jacobsthal numbers, sum formulas.

1. Introduction

Sequences have been fascinating topic for mathematicians for centuries. The Fibonacci and Lucas sequences are very well-known examples of second order recurrence sequences. The Fibonacci numbers are perhaps most famous for appearing in the rabbit breeding problem, introduced by Leonardo de Pisa in 1202 in his book called Liber Abaci. The Fibonacci sequences are a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence.

The sequence of Fibonacci numbers_fFng is de…ned by

Fn =Fn 1+Fn 2; n 2; F0= 0; F1= 1:

and the sequence of Lucas numbers_fLng is de…ned by

Ln =Ln 1+Ln 2; n 2; L0= 2; L1= 1:

The Fibonacci numbers, Lucas numbers and their generalizations have many interesting properties and applications to almost every …eld. In 1965, Horadam [8] de…ned a generalization of Fibonacci sequence, that

1

is, he de…ned a second-order linear recurrence sequence_fWn(W0; W1;r; s)g, or simplyfWng, as follows:

(1.1) Wn =rWn 1+sWn 2; W0=a; W1=b; (n 2)

whereW0; W1 are arbitrary complex numbers and r; sare real numbers, see also Horadam [7], [9] and [10]. Now these generalized Fibonacci numbers_fWn(a; b;r; s)g are also called Horadam numbers. The sequence

fWngn 0 can be extended to negative subscripts by de…ning

W n=

r

sW (n 1)+

1

sW (n 2)

forn= 1;2;3; ::: whens₆= 0:Therefore, recurrence (1.1) holds for all integern:

For some speci…c values ofa; b; rands, it is worth presenting these special Horadam numbers in a table as a speci…c name. In literature, for example, the following names and notations (see Table 1) are used for the special cases ofr; sand initial values.

Table 1. A few special case of generalized Fibonacci sequences. Name of sequence Notation: Wn(a; b;r; s) OEIS: [17]

Fibonacci Fn=Wn(0;1; 1;1) A000045

Lucas Ln=Wn(2;1; 1;1) A000032

Pell Pn=Wn(0;1; 2;1) A000129

Pell-Lucas Qn=Wn(2;2; 2;1) A002203

Jacobsthal Jn=Wn(0;1; 1;2) A001045

Jacobsthal-Lucas jn=Wn(2;1; 1;2) A014551

The evaluation of sums of powers of these sequences is a challenging issue. Two pretty examples are

n

X

i=1

Fi2=FnFn+1

and

n

X

i=1

Q2_i =1

2(QnQn+1 4):

In this work, we derive expressions for sums of second powers of generalized Fibonacci numbers. We present some works on sum formulas of powers of the numbers in the following Table 2.

Table 2. A few special study on sum formulas of second, third and arbitrary powers. Name of sequence sums of second powers sums of third powers sums of powers Generalized Fibonacci [1,2,6,11,12] [5,18] [3,4,13] Generalized Tribonacci [15]

Generalized Tetranacci [14,16]

Theorem 2.1. Forn 1 we have the following formulas: if(s+ 1) (r+s 1) (r s+ 1)₆= 0then

(a):

n

X

i=1

W_i2= (1 s)W

2

n+2+ (1 s r2 r2s)Wn2+1+ 2rsWn+1Wn+2+ (s 1)W12+s2(s 1)W02 2rsW1W0

(s+ 1) (r+s 1) (r s+ 1) :

(b):

n

X

i=1

Wi+1Wi=

rW_n2₊₂+rs2W_n2₊₁+ (1 r2 s2)Wn+1Wn+2 rW12 rs2W02+s( r2+s2 1)W1W0

(s+ 1) (r+s 1) (r s+ 1) :

Proof. Using the recurrence relation

Wn+2=rWn+1+sWn

i.e.

sWn =Wn+2 rWn+1

we obtain

s2Wn2 = Wn2+2+r2Wn2+1 2rWn+2Wn+1

s2W_n2 ₁ = W_n2₊₁+r2W_n2 2rWn+1Wn

s2W_n2 ₂ = W_n2+r2W_n2 ₁ 2rWnWn 1

s2W_n2 ₃ = W_n2 ₁+r2W_n2 ₂ 2rWn 1Wn 2

s2W_n2 ₄ = W_n2 ₂+r2W_n2 ₃ 2rWn 2Wn 3

.. .

s2W₂2 = W₄2+r2W₃2 2rW4W3

s2W₁2 = W₃2+r2W₂2 2rW3W2:

If we add the above equations by side by, we get

(2.1) s2

n

X

i=1 W_i2=

n_X+2

i=3

W_i2+r2

n+1 X

i=2

W_i2 2r

n_X+1

i=2

Wi+1Wi

Note that

n_X+2

i=3

W_i2 = W₁2 W₂2+W_n2₊₁+W_n2₊₂+ n

X

i=1 W_i2;

n_X+1

i=2

W_i2 = W₁2+W_n2₊₁+ n

X

i=1 W_i2;

n+1 X

i=2

Wi+1Wi = W2W1+Wn+2Wn+1+

n

X

i=1

We put them into the (2.1) we obtain

s2

n

X

i=1

W_i2 = ( W₁2 W₂2+W_n2₊₁+W_n2₊₂+ n

X

i=1

W_i2) +r2( W₁2+W_n2₊₁+ n

X

i=1 W_i2)

(2.2)

2r( W2W1+Wn+2Wn+1+

n

X

i=1

Wi+1Wi):

Next we calculatePn_i=1Wi+1Wi. Again, using the recurrence relation

Wn+2=rWn+1+sWn

i.e.

sWn =Wn+2 rWn+1

we obtain

sWn+1Wn = Wn+2Wn+1 rWn2+1

sWnWn 1 = Wn+1Wn rWn2

sWn 1Wn 2 = WnWn 1 rWn2 1 ..

.

sW3W2 = W4W3 rW₃2

sW2W1 = W3W2 rW22:

If we add the above equations by side by, we get

(2.3) s

n

X

i=1

Wi+1Wi = n_X+1

i=2

Wi+1Wi r n+1 X

i=2 W_i2

Note that

n+1 X

i=2

Wi+1Wi = W2W1+Wn+2Wn+1+

n

X

i=1

Wi+1Wi;

n_X+1

i=2

Wi2 = W12+Wn2+1+

n

X

i=1 Wi2:

If we put them into the (2.3) then it follows that

(2.4) s

n

X

i=1

Wi+1Wi = ( W2W1+Wn+2Wn+1+

n

X

i=1

Wi+1Wi) r( W12+Wn2+1+

n

X

i=1 W_i2):

Then, using

W2= (rW1+sW0)

and solving the system (2.2)-(2.4), the required results of (a) and (b) follow.

Takingr=s= 1in Theorem 2.1 (a) and (b), we obtain the following proposition.

(a): Pn_i=1W2

i = 12( 2W 2

n+1+ 2Wn+2Wn+1 2W1W0):

(b): Pn_i=1Wi+1Wi= ₂1(Wn2+2+Wn2+1 Wn+1Wn+2 W12 W02 W1W0):

From the above proposition, we have the following corollary which gives sum formulas of Fibonacci numbers (takeWn=Fn withF0= 0; F1= 1).

Corollary2.3. Forn 1; Fibonacci numbers have the following properties:

(a): Pn_i=1F_i2=1₂( 2Fn2+1+ 2Fn+2Fn+1):

(b): Pn_i=1Fi+1Fi= 1₂(Fn2+2+Fn2+1 Fn+1Fn+2 1):

Taking Wn = Ln with L0 = 2; L1 = 1 in the last proposition, we have the following corollary which

presents sum formulas of Lucas numbers.

Corollary2.4. Forn 1; Lucas numbers have the following properties:

(a): Pn_i=1L2

i = 12( 2L2n+1+ 2Ln+2Ln+1 4):

(b): Pn_i=1Li+1Li= 1₂(L2n+2+L2n+1 Ln+1Ln+2 7):

Takingr= 2; s= 1in Theorem 2.1 (a) and (b), we obtain the following proposition.

Proposition 2.5. If r= 2; s= 1 then forn 0 we have the following formulas:

(a): Pn_i=1W2

i = 12( 2Wn2+1+Wn+2Wn+1 W1W0):

(b): Pn_i=1Wi+1Wi= ₄1(Wn2+2+Wn2+1 2Wn+2Wn+1 W12 W02 2W1W0):

From the last proposition, we have the following corollary which gives sum formulas of Pell numbers (takeWn =Pn withP0= 0; P1= 1).

Corollary2.6. Forn 1; Pell numbers have the following properties:

(a): Pn_i=1P2

i =12( 2P 2

n+1+Pn+2Pn+1) =1₂PnPn+1:

(b): Pn_i=1Pi+1Pi= 1₄(Pn2+2+Pn2+1 2Pn+2Pn+1 1):

Taking Wn =Qn with Q0 = 2; Q1 = 2in the last proposition, we have the following corollary which presents sum formulas of Pell-Lucas numbers.

Corollary2.7. Forn 1; Pell-Lucas numbers have the following properties:

(a): Pn_i=1Q2

i = 12( 2Q 2

n+1+Qn+2Qn+1 4) = 1₂(QnQn+1 4):

(b): Pn_i=1Qi+1Qi= 1₄(Qn2+2+Q2n+1 2Qn+2Qn+1 16):

Ifr = 1; s= 2 then(s+ 1) (r+s 1) (r s+ 1) = 0 so we can’t use Theorem 2.1 directly. Therefore we need another method to …ndPn_i=1W2

i and

Pn

i=1Wi+1Wi which is given in the following theorem.

Theorem 2.8. If r= 1; s= 2 then forn 1 we have the following formulas:

(b): Pn_i=1Wi+1Wi= ₃₆1(5Wn2+2+ 4Wn2+1+ ( 9W12 20W02 20W1W0) 4 (W1 2W0)2n):

Proof.

(a): The proof will be by induction onn. Before the proof, we recall some information on generalized Jacobsthal numbers. A generalized Jacobsthal sequence_fWngn 0=fWn(W0; W1)gn 0 is de…ned by the second-order recurrence relations

(2.5) Wn=Wn 1+ 2Wn 2; W0=a; W1=b; (n 2)

with the initial values W0; W1 not all being zero. The sequence _fWngn 0 can be extended to negative subscripts by de…ning

W n = 1

2W (n 1)+ 1

2W (n 2)

for n = 1;2;3; :::: Therefore, recurrence (2.5) holds for all integer n: The …rst few generalized Jacobsthal numbers with positive subscript and negative subscript are given in the following Table 1.

Table 1. A few generalized Jacobsthal numbers

n Wn W n

0 W0

1 W1 1₂W0+1₂W1

2 2W0+W1 3₄W0 1₄W1

3 2W0+ 3W1 5₈W0+3₈W1

4 6W0+ 5W1 11₁₆W0 ₁₆5W1

5 10W0+ 11W1 21₃₂W0+11₃₂W1

6 22W0+ 21W1 43₆₄W0 21₆₄W1

Binet formula of generalized Jacobsthal sequence can be calculated using its characteristic equation which is given as

t2 t 2 = 0:

The roots of characteristic equation are

= 2; = 1

and the roots satisfy the following

+ = 1; = 2; = 3:

Using these roots and the recurrence relation, Binet formula can be given as

(2.6) Wn=

A n B n

= A 2

n _{B( 1)}n

3

We now prove (a) by induction onn. Ifn= 1wee that the sum formula reduces to the relation

(2.7) W12=

1 9(W

2

3 W22+W12 8W1W0):

Since

W2 = 2W0+W1;

W3 = 2W0+ 3W1;

(2.7) is true. Assume that the relation in (a) is true forn=m;i.e.,

m

X

i=1

W_i2= 1 9(W

2

m+2 Wm2+1 4 (W0+W1)W0+ (2W0 W1)2m):

Then we get

m_X+1

i=1

W_i2 = W_m2₊₁+ m

X

i=1 W_i2

= W_m2₊₁+1 9(W

2

m+2 Wm2+1 4 (W0+W1)W0+ (2W0 W1)2m)

= 1

9(W

2

m+2+ 8Wm2+1 4 (W0+W1)W0+ (2W0 W1)2m)

= 1

9(W

2

m+2+ 8Wm2+1 (2W0 W1)2 4 (W0+W1)W0+ (2W0 W1)2(m+ 1))

= 1

9(W

2

m+3 Wm2+2 4 (W0+W1)W0+ (2W0 W1) 2

(m+ 1))

= 1

9(W

2

(m+1)+2 W(2m+1)+1 4 (W0+W1)W0+ (2W0 W1) 2

(m+ 1))

where

(2.8) W_m2₊₂+ 8W_m2₊₁ (2W0 W1)2=Wm2+3 Wm2+2:

(2.8 ) can be proved by using Binet formula ofWn:Hence, the relation in (a) holds also forn=m+1:

(b): We now prove (b) by induction onn. Ifn= 1wee that the sum formula reduces to the relation (2.9) W2W1=

1 36(5W

2

3 + 4W22 13W12 36W02 4W1W0):

Since

W2 = 2W0+W1;

W3 = 2W0+ 3W1;

(2.9) is true. Assume that the relation in (b) is true forn=m;i.e.,

m

X

i=1

Wi+1Wi= 1 36(5W

2

m+2+ 4Wm2+1+ ( 9W12 20W02 20W1W0) 4 (W1 2W0) 2

Then we get

m_X+1

i=1

Wi+1Wi = Wm+2Wm+1+

m

X

i=1

Wi+1Wi

= 1

36(5W

2

m+2+ 4Wm2+1+ 36Wm+2Wm+1+ ( 9W12 20W02 20W1W0) 4 (W1 2W0)2m)

= 1

36(5W

2

m+2+ 4Wm2+1+ 36Wm+2Wm+1+ 4 (W1 2W0)2+ ( 9W12 20W02 20W1W0)

4 (W1 2W0)2(m+ 1))

= 1

36(5W

2

m+3+ 4Wm2+2+ ( 9W12 20W02 20W1W0) 4 (W1 2W0) 2

(m+ 1))

= 1

36(5W

2

(m+1)+2+ 4W(2m+1)+1+ ( 9W12 20W02 20W1W0) 4 (W1 2W0) 2

(m+ 1))

where

(2.10) 5Wm2+2+ 4Wm2+1+ 36Wm+2Wm+1+ 4 (W1 2W0)2= 5Wm2+3+ 4Wm2+2:

(2.10) can be proved by using Binet formula of Wn: Hence, the relation in (b) holds also for

n=m+ 1:

From the last theorem we have the following corollary which gives sum formulas of Jacobsthal numbers (takeWn =Jn withJ0= 0; J1= 1).

Corollary2.9. Forn 1; Jacobsthal numbers have the following property:

(a): Pn_i=1J_i2=1₉(J_n2₊₂ J_n2₊₁+n):

(b): Pn_i=1Ji+1Ji= ₃₆1(5Jn2+2+ 4Jn2+1 9 4n):

TakingWn=jn withj0= 2; j1= 1in the last theorem, we have the following corollary which presents

sum formulas of Jacobsthal-Lucas numbers.

Corollary2.10. Forn 1;Jacobsthal-Lucas numbers have the following property:

(a): Pn_i=1j_i2= 1₉(jn2+2 jn2+1 24 + 9n):

(b): Pn_i=1ji+1ji= ₃₆1(5jn2+2+ 4jn2+1 129 36n):

3. Summing Formulas of Generalized Fibonacci Numbers with Negative Subscripts The following theorem presents some summing formulas of generalized Fibonacci numbers with negative subscripts.

Theorem 3.1. Forn 1 we have the following formulas: If(s+ 1) (r+s 1) (r s+ 1)₆= 0 then

(a):

n

X

i=1 W2i=

(s 1)W2_n+1+ (r2+r2s+s 1)W2_n 2rsW n+1W n+ 2rsW1W0+ (1 s)W12

+(1 s r2 r2s)W₀2

(b):

n

X

i=1

W i+1W i=

rW2n+1 rs2W2n+ (r2+s2 1)W n+1W n+ (1 r2 s2)W1W0+rW12+rs2W02

(s+ 1) (r+s 1) (r s+ 1)

Proof. Using the recurrence relation

W n+2=rW n+1+sW n)W n =

r

sW n+1+

1

sW n+2

i.e.

sW n=W n+2 rW n+1

we obtain

s2W2_n = W2_n+2+r2W2_n+1 2rW n+2W n+1

s2W2n+1 = W2n+3+r2W2n+2 2rW n+3W n+2

s2W2_n+2 = W2_n+4+r2W2_n+3 2rW n+4W n+3

s2W2_n+3 = W2_n+5+r2W2_n+4 2rW n+5W n+4

.. .

s2W2₃ = W2₁+r2W2₂ 2rW 1W 2

s2W2₂ = W₀2+r2W2₁ 2rW0W 1

s2W2₁ = W₁2+r2W₀2 2rW1W0:

If we add the above equations by side by, we get

s2

n

X

i=1

W2_i = (W₁2+W₀2 W2_n+1 W2_n+ n

X

i=1

W2_i) +r2(W₀2 W2_n+ n

X

i=1 W2_i)

(3.1)

2r(W1W0 W n+1W n + n

X

i=1

W i+1W i)

Next we calculatePn_i=1W i+1W i:Again using the recurrence relation

W n+2=rW n+1+sW n)W n =

r

sW n+1+

1

sW n+2

i.e.

sW n=W n+2 rW n+1

sW n+1W n = W n+2W n+1 rW2n+1

sW n+2W n+1 = W n+3W n+2 rW2n+2

sW n+3W n+2 = W n+4W n+3 rW2n+3

sW n+4W n+3 = W n+5W n+4 rW2n+4 ..

.

sW 2W 3 = W 1W 2 rW22

sW 1W 2 = W0W 1 rW21

sW0W 1 = W1W0 rW02:

If we add the above equations by side by, we get

(3.2) s

n

X

i=1

W i+1W i = (W1W0 W n+1W n+ n

X

i=1

W i+1W i) r(W02 W2n+ n

X

i=1 W2_i)

Then, solving the system (3.1)-(3.2), the required results of (a) and (b) follow.

Takingr=s= 1in Theorem 3.1 (a) and (b), we obtain the following proposition.

Proposition 3.2. If r=s= 1 then forn 1 we have the following formulas:

(a): Pn_i=1W2

i=12(2W 2

n 2W n+1W n+ 2W1W0 2W02):

(b): Pn_i=1W i+1W i=1₂( W2n+1 W2n+W n+1W n W1W0+W12+W02):

From the above proposition, we have the following corollary which gives sum formulas of Fibonacci numbers (takeWn=Fn withF0= 0; F1= 1).

Corollary3.3. Forn 1; Fibonacci numbers have the following properties.

(a): Pn_i=1F2

i= 12(2F2n 2F n+1F n):

(b): Pn_i=1F i+1F i=1₂( F2n+1 F2n+F n+1F n+ 1):

Taking Wn = Ln with L0 = 2; L1 = 1 in the last proposition, we have the following corollary which

presents sum formulas of Lucas numbers.

Corollary3.4. Forn 1; Lucas numbers have the following properties.

(a): Pn_i=1L2

i= 12(2L2n 2L n+1L n 4):

(b): Pn_i=1L i+1L i=1₂( L2n+1 L2n+L n+1L n+ 3):

Takingr= 2; s= 1in Theorem 3.1 (a) and (b), we obtain the following proposition.

Proposition 3.5. If r= 2; s= 1 then forn 1 we have the following formulas:

(b): Pn_i=1W i+1W i=1₄( W2n+1 W2n+ 2W n+1W n+W12+W02 2W1W0):

From the last proposition, we have the following corollary which gives sum formulas of Pell numbers (takeWn =Pn withP0= 0; P1= 1).

Corollary3.6. Forn 1; Pell numbers have the following properties.

(a): Pn_i=1P2_i= ₂1(2P2n P n+1P n):

(b): Pn_i=1P i+1P i =1₄( P2n+1 P2n+ 2P n+1P n+ 1):

Taking Wn =Qn with Q0 = 2; Q1 = 2in the last proposition, we have the following corollary which presents sum formulas of Pell-Lucas numbers.

Corollary3.7. Forn 1; Pell-Lucas numbers have the following properties.

(a): Pn_i=1Q2_i =₂1(2Q2_n Q n+1Q n 4):

(b): Pn_i=1Q i+1Q i= 1₄( Q2n+1 Q2n+ 2Q n+1Q n):

Ifr = 1; s= 2 then(s+ 1) (r+s 1) (r s+ 1) = 0 so we can’t use Theorem 3.1 directly. Therefore we need another method to …ndPn_i=1W2

i and

Pn

i=1W i+1W i which is given in the following theorem.

Theorem 3.8. If r= 1; s= 2 then forn 1 we have the following formulas:

(a): Pn_i=1W2_i=1₉( W2_n+1+W2_n+ (W₁2 W₀2) + (W1 2W0)2n):

(b): Pn_i=1W i+1W i=₂₇1( 2W2n+1+4W2n 7W n+1W n+(W1+ 4W0) (2W1 W0) 3 (W1 2W0)2n):

Proof. (a) and (b) can be proved by mathematical induction.

(a): The proof will be by induction onn. Ifn= 1wee that the sum formula reduces to the relation

(3.3) W2₁=1

9(2W

2

0 4W0W1+ 2W12+W21)

:

Since

W 1= (

1 2W0+

1 2W1)

(3.3) is true. Assume that the relation in (a) is true forn=m;i.e.

m

X

i=1 W2i=

1 9( W

2

m+1+W2m+ (W12 W02) + (W1 2W0) 2

Then we get

m_X+1

i=1

W2_i = W2_(m+1)+ m

X

i=1 W2_i

= W2_{m 1}+1 9( W

2

m+1+W2m+ (W12 W02) + (W1 2W0)2m)

= 1

9( W

2

m+1+W2m+ 9W2m 1+ (W12 W02) + (W1 2W0)2m)

= 1

9( W

2

m+1+W2m+ 9W2m 1 (W1 2W0) 2

+ (W12 W02) + (W1 2W0) 2

(m+ 1))

= 1

9( W

2

m+W2m 1+ (W12 W02) + (W1 2W0) 2

(m+ 1))

= 1

9( W

2

(m+1)+1+W2(m+1)+ (W12 W02) + (W1 2W0) 2

(m+ 1))

where

(3.4) W2_m+1+W2_m+ 9W2_{m 1} (W1 2W0)2= W2m+W2m 1:

(3.4) can be proved by using Binet formula ofWn:Hence, the relation in (a) holds also forn=m+1:

(b): We now prove (b) by induction on n. If n = 1 wee see that the sum formula reduces to the relation

(3.5) W0W 1=

1 27( W

2

1 18W02+ 4W21+ 19W0W1 7W0W 1):

Since

W 1= (

1 2W0+

1 2W1);

(3.5) is true. Assume that the relation in (b) is true forn=mi.e.,

m

X

i=1

W i+1W i= 1 27( 2W

2

Then we get

m_X+1

i=1

W i+1W i = W (m+1)+1W (m+1)+

m

X

i=1

W i+1W i

= W mW m 1+

1 27( 2W

2

m+1+ 4W2m 7W m+1W m

+ (W1+ 4W0) (2W1 W0) 3 (W1 2W0)2m)

= 1

27( 2W

2

m+1+ 4W2m 7W m+1W m+ 27W mW m 1

+ (W1+ 4W0) (2W1 W0) 3 (W1 2W0)2m)

= 1

27( 2W

2

m+1+ 4W2m 7W m+1W m+ 27W mW m 1+ 3 (W1 2W0)2

+ (W1+ 4W0) (2W1 W0) 3 (W1 2W0)2(m+ 1))

= 1

27( 2W

2

m+ 4W2m 1 7W mW m 1+ (W1+ 4W0) (2W1 W0)

3 (W1 2W0)2(m+ 1))

= 1

27( 2W

2

(m+1)+1+ 4W2(m+1) 7W (m+1)+1W (m+1)+ (W1+ 4W0) (2W1 W0)

3 (W1 2W0)2(m+ 1))

where (3.6)

2W2_m+1+4W2_m 7W m+1W m+27W mW m 1+3 (W1 2W0)2= 2W2m+4W2m 1 7W mW m 1:

(3.6) can be proved by using Binet formula ofWn:Hence, the relation in (b) holds also forn=m+1:

From the last theorem, we have the following corollary which gives sum formula of Jacobsthal numbers (takeWn =Jn withJ0= 0; J1= 1).

Corollary3.9. Forn 1; Jacobsthal numbers have the following property:

(a): Pn_i=1J2_i= 1₉( J2_n+1+J2_n+ 1 +n):

(b): Pn_i=1J i+1J i= ₂₇1( 2J2n+1+ 4J2n 7J n+1J n+ 2 3n):

Taking Wn = jn with j0 = 2; j1 = 1 in the last proposition, we have the following corollary which presents sum formulas of Jacobsthal-Lucas numbers.

Corollary3.10. Forn 1;Jacobsthal-Lucas numbers have the following property:

(a): Pn_i=1j2_i= 1₉( j2n+1+j2n 3 + 9n):

(b): Pn_i=1j i+1j i=₂₇1( 2j2n+1+ 4j2n 7j n+1j n 27n):

4. Conclusion

recurrence sequences, too. We have written sum identities in terms of the generalized Fibonacci sequence, and then we have presented the formulas as special cases the corresponding identity for the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. All the listed identities in the corollaries may be proved by induction, but that method of proof gives no clue about their discovery. We give the proofs to indicate how these identities, in general, were discovered.

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