Sums of Squares of Jacobsthal Numbers
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The document presents various identities related to the sums of squares of Jacobsthal numbers using Binet forms. It includes several propositions demonstrating equalities for these sums, backed by proofs. The paper discusses both Jacobsthal and Jacobsthal-Lucas sequences and provides references for further reading.
![IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 2015), PP 62-64
www.iosrjournals.org
DOI: 10.9790/5728-11646264 www.iosrjournals.org 62 | Page
Sums of Squares of Jacobsthal Numbers
A.Gnanam1
, B.Anitha2
1
(Assistant Professor,Department of Mathematics,Government Arts College-Trichy-22,India)
2
(Research Scholar,Department of Mathematics,Government Arts College-Trichy-22,India)
Abstract: There are many identities on Jacobsthal sequence of numbers. Here we try tofind some more
identities on Sums of Squares of Consecutive Jacobsthal numbers using Binet forms.
Keywords: Jacobsthal numbers, sums of squares.
I. Introduction
The Jacobsthal and Jacobsthal –Lucas sequences and are defined by the recurrence relations
for ---------------(1)
for ---------------(2)
Applications of these two sequences to curves are given in [1]. Sequence (1) appears in [2] but (2)
does not. From (1) and (2) we thus have the following tabulation for the Jacobsthal numbers and the
Jacobsthal –Lucas sequences .
n 0 1 2 3 4 5 6 7 8 9 10 …
0 1 1 3 5 11 21 43 85 171 341 …
2 1 5 7 17 31 65 127 257 511 1025 …
When required we can extend these sequences through negative values of n by means of the recurrence
(1) and (2). Observe that all the and except are odd by virtue of the definitions.
In [3], the Binet forms of Jacobsthal form are given as
Considering Binet forms, Jacobsthal sequences can also be represented as
=
Based on this construction, we obtain some identities.
Proposition 1.
For every the following equality holds
Proof.
Remark:
For every the following equality holds](https://image.slidesharecdn.com/h011646264-160129081222/85/Sums-of-Squares-of-Jacobsthal-Numbers-1-320.jpg)
![ Sums of Squares of Jacobsthal Numbers
DOI: 10.9790/5728-11646264 www.iosrjournals.org 64 | Page
Proposition 6.
For every the following equality holds
Proof.
By the formula we have
Proposition 7.
For every the following equality holds
.
Proof.
.
Proposition 8.
For every the following equality holds
Proof .
References
[1] A.F.Horadam, “Jacobsthal and pell curves” The Quartely 26.I(1988):79-83.
[2] N.J.A.Sloane, A Handbook of integer Sequences.NewYork:Academic Press,1973.
[3] A.F.Horadam, Jacobsthal Representation Numbers,Unversity of New England,Armidale,2351,Australia.
[4] Zvonko Cerin, Sums of Squares and Products of Jacobsthal Numbers,Journal of integer Sequences,Vol. 10 (2007)](https://image.slidesharecdn.com/h011646264-160129081222/85/Sums-of-Squares-of-Jacobsthal-Numbers-3-320.jpg)
Sums of Squares of Jacobsthal Numbers
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 2015), PP 62-64 www.iosrjournals.org DOI: 10.9790/5728-11646264 www.iosrjournals.org 62 | Page Sums of Squares of Jacobsthal Numbers A.Gnanam1 , B.Anitha2 1 (Assistant Professor,Department of Mathematics,Government Arts College-Trichy-22,India) 2 (Research Scholar,Department of Mathematics,Government Arts College-Trichy-22,India) Abstract: There are many identities on Jacobsthal sequence of numbers. Here we try tofind some more identities on Sums of Squares of Consecutive Jacobsthal numbers using Binet forms. Keywords: Jacobsthal numbers, sums of squares. I. Introduction The Jacobsthal and Jacobsthal –Lucas sequences and are defined by the recurrence relations for ---------------(1) for ---------------(2) Applications of these two sequences to curves are given in [1]. Sequence (1) appears in [2] but (2) does not. From (1) and (2) we thus have the following tabulation for the Jacobsthal numbers and the Jacobsthal –Lucas sequences . n 0 1 2 3 4 5 6 7 8 9 10 … 0 1 1 3 5 11 21 43 85 171 341 … 2 1 5 7 17 31 65 127 257 511 1025 … When required we can extend these sequences through negative values of n by means of the recurrence (1) and (2). Observe that all the and except are odd by virtue of the definitions. In [3], the Binet forms of Jacobsthal form are given as Considering Binet forms, Jacobsthal sequences can also be represented as = Based on this construction, we obtain some identities. Proposition 1. For every the following equality holds Proof. Remark: For every the following equality holds
- 2. Sums of Squares of Jacobsthal Numbers DOI: 10.9790/5728-11646264 www.iosrjournals.org 63 | Page Proposition 2. For every Proof. Proposition 3: For every Proof . Proposition 4. For every Proof . By the formula we have Proposition 5. For every , the following equality holds Proof. By the formula we have
- 3. Sums of Squares of Jacobsthal Numbers DOI: 10.9790/5728-11646264 www.iosrjournals.org 64 | Page Proposition 6. For every the following equality holds Proof. By the formula we have Proposition 7. For every the following equality holds . Proof. . Proposition 8. For every the following equality holds Proof . References [1] A.F.Horadam, “Jacobsthal and pell curves” The Quartely 26.I(1988):79-83. [2] N.J.A.Sloane, A Handbook of integer Sequences.NewYork:Academic Press,1973. [3] A.F.Horadam, Jacobsthal Representation Numbers,Unversity of New England,Armidale,2351,Australia. [4] Zvonko Cerin, Sums of Squares and Products of Jacobsthal Numbers,Journal of integer Sequences,Vol. 10 (2007)