A MULTISET-VALUED FIBONACCI-TYPE SEQUENCE

Transcription

1 Avces Applctos Dscrete Mthemtcs Volume Number 008 Pes Publshe Ole: Mrch Ths pper s vlble ole t Pushp Publsh House A MULTISET-VALUED IBONACCI-TYPE SEQUENCE TAMÁS KALMÁR-NAGY Deprtmet of Aerospce Eeer Tes A & M Uversty Collee Stto TX 7785 U S A e-ml: scretemth@klmrycom Abstrct We vestte the structure of multset-vlue bocc-type sequece crete by set uo/sum/fferece opertos A close-form eert fucto s erve to eplctly chrcterze the elemets of these sets ther ect veres vrces The eometrc me of the sets s umerclly emostrte to epoetlly row we prove tht the epoet s the squre root of the ole rto Itroucto Oe of the smplest emples of rom mtr proucts s the soclle rom bocc seres (Vswth [0]) Ths sequece s eerte by the rom seco-orer fferece equto ± 0 () where the th term s ether the sum or fferece of the prevous two terms ( or re pcke epeetly wth probblty / t ech step) The structure rowth propertes of the rom bocc sequece hve bee stue by severl uthors (Embree Trefethe [5]; Sre Krpvsky [9]; Ch [3 ]; Jvresse et l [6]; Klmár-Ny [7]; Mkover McGow [8]; B []) Vswth [0] utlze powerful 000 Mthemtcs Subect Clssfcto: B39 05A5 Keywors phrses: bocc sequece eert fucto multset Mkowsk sum Receve November 008

2 86 TAMÁS KALMÁR-NAGY combto of rom mtr theory tervl rthmetc to show tht the sequece lmost surely veres wth epoet of 3 e ν lm 3 Embree Trefethe [5] umerclly vestte ± β eerlzto of the rom bocc sequece The mesure of verece for the rom bocc seres c be epresse s pth vere by coser ll the possble vlues c tt Let h eote the multset of possble - ot ecessrly fferet - vlues of A multset s collecto of obects whch elemets my occur more th oce (Blzr []) The epoet (Vswth s umber) ν c the be formlly epresse s the lmt ( m) m ν lm lm () 0 h 0 h where the vere (eometrc me) s tke over ll the m possble vlues of t step (here m ) Eve thouh there s stro correlto betwee subsequet elemets y prtculr relzto of ths rom sequece the sme epoet ν woul chrcterze the rom sequece whose th elemet s romly chose from h Ths observto proves the motvto for ths stuy Whle recurso relto for the eert fucto of h c be costructe the ol of ths pper s to prove eve smpler emple of multset-vlue recurrece to erve some of ts bsc propertes A Multset-vlue Recurrece Motvte by stues o the rowth of rom bocc sequeces here we trouce multset-vlue bocc-type recurrece Strt wth the two sets f f { } the set f 3 s costructe s the uo of the Mkowsk sum fferece of the prevous two sets ( f f ) ( f f ) { 0 } (3) f3

3 A MULTISET-VALUED IBONACCI-TYPE SEQUENCE 87 where eote the Mkowsk sum fferece of two sets X Y efe s X Y { y X y Y} () X Y { y X y Y} X ( Y ) (5) I eerl we efe f ( f f ) ( f f ) (6) Notce tht f s multset e elemets my occur more th oce The umber of tmes elemet occurs multset s clle ts multplcty (Blzr []) To vo cofuso ste of the str otto for multsets here ( ) eotes the umber of occurreces of elemet The frst few f s re ve by f f {} f 3 { 0 } (7) { 3} { ( ) 3} f { 3 ( ) ( 6) 3( ) 5} f 5 f { 6 ( 7) ( ) 0( 35) ( 35) ( ) 6( 7) 8} (8) 6 ew turl questos rse: Wht ectly re the elemets of f? How my elemets re there f? Wht re vrous sttstcl propertes (me vrce eometrc me) of f? It s the ol of the et secto to swer these questos 3 Geert ucto Sttstcl Propertes of f rst we chrcterze the etreml elemets ( thus the re) of f I the follow eotes the -th bocc umber

4 88 TAMÁS KALMÁR-NAGY Theorem m m f f Proof The frst sttemet follows esly from m f m f m f m f m f The m f m f m f m f The propose soluto m f stsfes ths equto (becuse of the etty ) s well s the tl cotos m f m f f The et theorem proves smple epresso for the crlty of Theorem f Proof The crlty of f s epresse s f ( f f ) ( f f ) f f f f f f Clerly f stsfes ths recurrece the tl cotos f f To compute the frequeces of the vrous teers pper f we use the eert fucto pproch (see the ecellet tretse by Wlf []) I our problem forml power seres ( ) (9) wll ecoe formto bout f I prtculr the e wll correspo to the teer the set f the coeffcet specfes the frequecy (umber of occurreces) of ths teer f Ths proves oe-to-oe mpp betwee the set f the eert fucto ( ) or emple: 3 f { ( ) 3} ( ) f 5 { 3 ( ) ( 6) 3( ) 5} ( 5 ) 6 (0) If the eert fucto of f s ( ) tht of f s obvously

5 A MULTISET-VALUED IBONACCI-TYPE SEQUENCE 89 ( ) The uo Mkowsk sum/fferece of two sets c be esly trslte to opertos betwee ther eert fuctos s f f ( ) ( ) () f f ( ) ( ) () f ( ) ( ) f f f (3) We re ow the posto to epress the eert fucto of f terms of those for f f Recll tht f ( f f ) ( f f ) thus ( ) ( ) ( ) ( ) ( ) ( ) () The tl cotos f f { } specfy those o the eert fucto ( ) ( ) (5) A ce close-form epresso c be fou for ( ): Theorem 3 ( ) ( ) (6) Proof Smple substtuto cofrms tht ths fucto stsfes the tl cotos (5) We wll ow show tht Eq (6) s soluto of the recurrece equto () ( ) ( ) The left h se s smplfe s ( ) (7) ( ) ( ) ( ) ( ) ( ) (8)

6 TAMÁS KALMÁR-NAGY 90 To evlute the rht h se of Eq (7) we frst clculte ( ) (9) The ( ) ( ) ( ) ( ) ( ) ( ) (0) The proof s complete by utlz the etty Corollry The eert fucto ( ) stsfes the fuctol equto ( ) () We c use the eert fucto formlsm to cofrm the erler result o the crlty of f The sze of f s smply the totl umber of the vrous teers t cots ths umber s smply the sum of coeffcets of the eert fucto ( ) f () The eert fucto ( ) c be wrtte s the boml epso ( ) 0 (3) Ths proves eplct escrpto of the set : f the elemets re wth frequecy [ ] 0 Note tht ths rely proves other proof of Theorem

7 A MULTISET-VALUED IBONACCI-TYPE SEQUENCE 9 ew elemetry sttstcl propertes of f re supple by Theorem 5 The vere µ vrce ν of the elemets f re ve by µ () ν (5) Proof The vere vrce c be compute rectly from the eert fucto ( ) (Wlf []) I prtculr µ ( ) ( ) (6) ν ( ) ( ) (7) Dfferetto of the eert fucto wrt yels ( ) ( ) (( ) ) (8) therefore ( ) Sce ( ) µ The rthmc ervtves re compute s ( ) 3 ( ) ( ) (9) ( ) ( ) 6 3 ( ) ( ) (30) therefore ν lly we coser the eometrc me of f (more precsely tht of the bsolute vlues of ts elemets)

8 9 TAMÁS KALMÁR-NAGY ( f ) G( f ) (3) 0 f where f s the umber of elemets f Tble shows the vlues of the eometrc me for vrous vlues of ther rthms Tble Growth of the eometrc me of f G ( f ) l G( f ) Bse o Tble the rowth of l G( f ) s ler wth slope of ppromtely 0 therefore the eometrc me of f scles wth epoet of κ ep( 0) 7 e κ lm G ( f ) 7 (3) I other wors the (rom) sequece whose th elemet s romly pcke from f ehbts epoetl rowth chrcterze by κ Our ppromto of κ s very close to tht of the squre root of the ole 5 rto φ φ 7 the error s ust 005% Iee Theorem 6 κ lm G ( f ) φ (33) Proof Itrouc the eert fucto (3) c lso be wrtte s

9 A MULTISET-VALUED IBONACCI-TYPE SEQUENCE 93 ( ) (3) The epoets of the eert fuctos re the elemets of f the coeffcets correspo to ther frequeces Thus the eometrc me of f s compute s ( ) f G ep (35) Sce the summ (35) escrbes wehte boml strbuto I the lmt the boml strbuto c be well ppromte by the cotuous orml strbuto (Wesste []) ep ~ π (36) The sum s ppromte s ~ π ep ~ π π 0 ep ep ~ (37) Sce ( ) γ π 0 ep (38)

10 9 TAMÁS KALMÁR-NAGY we obt γ G ( f ) ~ (39) Here γ s the Euler-Mschero costt lly G( f ) ~ ~ φ ~ φ φ (0) thus lm G( f ) e φ κ φ () Coclusos Motvte by stues o the rom bocc sequece we efe vestte recurrece whose elemets re multsets A closeform eert fucto s erve to eplctly chrcterze the elemets of these multsets ther ect veres vrces The eometrc me of the multsets s prove to row epoetlly scl wth the squre root of the ole rto Ackowleemet The uthor woul lke to ckowlee the help of Professor Péter Vlkó cret Tble Refereces [] Z-Q B O the cycle epso for the Lypuov epoet of prouct of rom mtrces J Phys A: Mth Theor 0 (007) [] W D Blzr Multset theory Notre Dme J orml Loc 30() (989) [3] H Ch The symptotc rowth rte of rom bocc type sequeces I bocc Qurterly 3(3) (005) 3-55

11 A MULTISET-VALUED IBONACCI-TYPE SEQUENCE 95 [] H Ch The symptotc rowth rte of rom bocc type sequeces II bocc Qurterly () (006) 73-8 [5] M Embree L N Trefethe Growth ecy of rom bocc sequeces Procees of the Royl Socety Loo Seres A 55(987) (999) 7-85 [6] E Jvresse B Rttu T De L Rue Arv preprt mth PR/06860 (006) [7] T Klmár-Ny The rom bocc recurrece the vsble pots of the ple J Phys A: Mth Ge 39(0) (006) L33-L38 [8] E Mkover J McGow A elemetry proof tht rom bocc sequeces row epoetlly J Number Theory () (006) 0- [9] C Sre P L Krpvsky Rom bocc sequeces J Phys A: Mth Ge 3 (00) [0] D Vswth Rom bocc sequeces the umber 3988 Mth Comp 69(3) (000) 3-55 [] E W Wesste Boml Dstrbuto [] H S Wlf Geertfuctooy Acemc Press Bosto 990