On the k-lucas Numbers of Arithmetic Indexes
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1 Alied Mthetics htt://d.doi.og/0.436/ Published Olie Octobe 0 (htt:// O the -ucs Nubes of Aithetic Idees Segio lco Detet of Mthetics d Istitute fo Alied Micoelectoics (IUMA) Uivesity of s Pls de G Ci s Pls Si Eil: sflco@d.ulgc.es Received Juy 3 0; evised Setebe 0; cceted Jue 0 ABSTRACT I this e we study the -ucs ubes of ithetic idees of the fo + whee is tul ube d is less th. We ove foul fo the su of these ubes d ticully the sus of the fist -ucs ubes d the fo the eve d the odd -ucs ubes. te we fid the geetig fuctio of these ubes. Below we ove these se fouls fo the lteted -ucs ubes. The we ove eltio betwee the -ibocci ubes of idees of the fo d the -ucs ubes of idees ultile of 4. illy we fid foul fo the su of the sque of the -ibocci eve ubes by e of the -ucs ubes. Keywods: -ibocci Nubes; -ucs Nubes; Geetig uctio. Itoductio et us eebe the -ucs ubes e defied [] by the ecuece eltio with the iitil coditios 0 Aog othe oeties the Biet Idetity estblishes 4 beig d 4 the chcteistic oots of the ecu- ece equtio 0. Evidetly 4. Moeove it is veified [ Theoe.4] tht. If we ly itetively the equtio the we will fid foul tht eltes the ucs ubes to the ibocci ubes: (.) This foul is siil to the Covolutio foul fo the ibocci ubes [3]. Moeove we defie. The if we do = i oul (.) obti.. O the -ucs Nubes of Aithetic Ide We begi this sectio with foul tht eltes ech othe soe -ucs ubes... Theoe (The -ucs Nubes of Aithetic Ide) If is oull tul ube d = 0... the (.) Poof. I [4] it is oved. The If = 0 the I this cse if = + the odd -ucs ube c be eessed i the fo Alyig itetively oul (.) the geel te fo 0 c be witte lie o-lie cobitio of the fo 0 I ticul if = the 0 Coyight 0 SciRes.
2 S. ACON 03.. Geetig uctio of the Sequece { + } et l ; be the geetig fuctio of the sequece. Tht is l ; d The l ; fo whee 3 3 3? l 3 l ; o oe to te ito ccout oul (.). So the geetig fuctio of the sequece is l ;. As ticul cse if = the = 0 d the geet- ig fuctio of the -ucs sequece is l tht fo the clssicl ucs se- quece is l If we wt to te out the two bisectio sequeces of the clssicl ucs sequece ( = ) the esective geetig fuctios e = d = 0: 3 l;0 tht geetes the sequece tht geetes = d = : l; the sequece.3. Theoe (Su of the -ucs Nubes of Aithetic Ide) If is oull tul ube d = 0... the 0 Poof. (.) 0 0 becuse ( ) d fte lyig the foul fo the su of geoetic ogessio. tht fo the clssicl ucs ubes is 0.4. Coolly (Su of Cosecutive Odd -ucs Nubes) If = 0 d = + Equtio (.) is 0 I this cse the su of the fist -ucs ubes is (fo = 0) (.3) 0.. Coolly (Su of Cosecutive Eve -ucs Nubes) If = 0 d = the Equtio (.) is (.4) 0 I this cse if = we obti the foul fo the su of the fist eve -ucs ubes 0 Coyight 0 SciRes.
3 04 S. ACON d fo the clssicl ucs ubes is Theoe 3 (Su of Alteted -ucs Nubes of Aithetic Ide) o > 0 d = 0... the su of lteted -ucs ubes is ( ) Poof. As i the evious theoe 0 0 ( ) ( ) ( ) ( ) ( ).7. Coolly 3 (Su of Cosecutive Odd -ucs Nubes) As ticul cse if = + d = 0 0 The fo = 0 we obti the su ted -ucs ubes 0 the clssicl ucs ubes is 0 3. Alteted of the fist lte- tht fo.8. Coolly 4 (Su of Cosecutive Alteted Eve -ucs Nubes) If = 0 d = + the 0 Ad fo the fist cosecutive ubes lteted eve -ucs 0 4 tht fo the clssicl ucs ubes is O the -ibocci Nubes of Idees d the -ucs Nubes I this sectio we will study eltio betwee the ubes d. 3.. Theoe 4 (A Reltio betwee Soe -ibocci d the -ucs Nubes) o it is Poof. (3.) 0 0 Coyight 0 SciRes.
4 S. ACON 0 I ticul if = it is Tig ito ccout if we ed o ul (3.) we fid tht this foul c be eessed s o tht is the se The lyig oul (.) to the secod hd ight of this equtio with = 4 d = 3 fo the fist te d = fo the secod (3.) 4 3 ( ) 3 4 We ty to silify the secod hd ight of this equtio. o tht we will ove the followig e. 3.. e 4 (3) 4 Poof. We will ly the followig fouls: The: (eltio) (egtive) (covolutio) b b b (defiitio) ( 4) ( 4) ( 4) ( ) () ( ) () ( ) ( ) ( ) ( ) (by eltio) (by covolutio) (by egtive) ( ) 4 ( ) (by defiitio) Ad lyig this e to Equtio (3.) we will hve: tht is 4 fo whee 4 ( ) If i Equtio (3.3) it is = 0 the it is 4 d lyig the ouls (.) d 4 (.4) Tht is I ticul fo the clssicl ucs ubes ( = ) it 43 4 is 0. Coyight 0 SciRes.
5 06 S. ACON 4. Acowledgeets This wo hs bee suoted i t by CICYT Poect ube MTM C03-0 fo Miisteio de Educció y Cieci of Si. REERENCES [] S. lco O the -ucs Nubes Itetiol Joul of Coteoy Mtheticl Scieces Vol. 6 No [] S. lco d A. Plz O the ibocci -Nubes Chos Solitos & ctls Vol. 3 No doi:0.06/.ch os [3] S. lco d A. Plz The -ibocci Sequece d the Pscl -Tigle Chos Solitos & ctls Vol. 33 No doi:0.06/.chos [4] S. lco d A. Plz O -ibocci Nubes of Aithetic Idees Alied Mthetics d Couttio Vol doi:0.06/.c Coyight 0 SciRes.
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