Về phương trình tuyến tính với các số fibonacci

ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - ĐINH THỊ HUYỀN VỀ PHƢƠNG TRÌNH TUYẾN TÍNH VỚI CÁC SỐ FIBONACCI Chun ngành: Phƣơng pháp Tốn sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS NƠNG QUỐC CHINH THÁI NGUN - 2019 ✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▼ð ✤➛✉ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✷ ✹ ✷ ❈→❝ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ sè ❋✐❜♦♥❛❝❝✐ ✾ ✶✳✶ ✶✳✷ ✶✳✸ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ❉➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ✼✼✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ✽✵✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✱ ❝→❝ ❦❤→✐ ♥✐➺♠ ✳ ✳ ❚r÷í♥❣ ❤đ♣ m = ✈➔ m = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ x(i) < b, ✈ỵ✐ ♠å✐ i ✳ ✳ ✳ ✳ ✳ ✳ rữớ ủ tỗ t i x(i) ≥ b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❦➳t q✉↔ ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ t➟♣ S1 ✳ ✳ ✳ ✳ ✳ ❚r÷í♥❣ ❤đ♣ b ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ù♥❣ ỵ ỵ t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✻ ✼ ✾ ✶✸ ✶✻ ✷✷ ✷✺ ✷✾ ✸✸ ✸✻ ✸✽ ✸✾ ✶ ▲í✐ ❝↔♠ ì♥ rữợ t tổ ỷ t ỡ t P ổ ố ữợ tổ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❑❤✐ ❜➢t ✤➛✉ ♥❤➟♥ ✤➲ t➔✐ t❤ü❝ sü tæ✐ ❝↔♠ ♥❤➟♥ ✤➲ t➔✐ ♠❛♥❣ ♥❤✐➲✉ ♥ë✐ ❞✉♥❣ ♠ỵ✐ ♠➫✳ ❍ì♥ ♥ú❛ ✈ỵ✐ ✈è♥ ❦✐➳♥ t❤ù❝ ➼t ä✐ ♥➯♥ r➜t ❦❤â ✤➸ t✐➳♣ ❝➟♥ ✤➲ t➔✐✳ ▼➦❝ ❞ị r➜t ❜➟♥ rë♥ tr♦♥❣ ❝ỉ♥❣ ✈✐➺❝ ♥❤÷♥❣ ❚❤➛② ✈➝♥ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ✈➔ t➙♠ ❤✉②➳t tr♦♥❣ ữợ tổ tr sốt t❤í✐ ❣✐❛♥ tỉ✐ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✳ ❚r♦♥❣ q✉→ tr➻♥❤ t✐➳♣ ❝➟♥ ✤➲ t➔✐ ✤➳♥ q✉→ tr➻♥❤ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ ❚❤➛② ❧✉æ♥ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❈❤♦ ✤➳♥ ❜➙② ❣✐í ❧✉➟♥ ✈➠♥ t❤↕❝ s➽ ❝õ❛ tỉ✐ ✤➣ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤✱ ①✐♥ ❝↔♠ ì♥ ❚❤➛② ✤➣ ✤ỉ♥ ✤è❝ ♥❤➢❝ ♥❤ð tỉ✐✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❚♦→♥ ✲ ❚✐♥ ✈➔ P❤á♥❣ ✣➔♦ t↕♦ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚ỉ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❝→❝ ❚❤➛②✱ ❈ỉ t t tr t ỳ tự qỵ ❝ơ♥❣ ♥❤÷ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ♥❤➜t ✤➸ tæ✐ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚æ✐ ①✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ tr÷í♥❣ ❚❍P❚ ❍♦❛ ▲÷ ❆ ✲ ◆✐♥❤ ❇➻♥❤ ♥ì✐ tỉ✐ ❝ỉ♥❣ t→❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❣✐ó♣ ✤ï tỉ✐ ❤♦➔♥ t❤➔♥❤ ❝ỉ♥❣ ✈✐➺❝ ❝❤✉②➯♥ ♠ỉ♥ t↕✐ ♥❤➔ tr÷í♥❣ ✤➸ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❝❤÷ì♥❣ tr➻♥❤ ❤å❝ t➟♣ ❝❛♦ ❤å❝✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❤ú♥❣ ♥❣÷í✐ ❦❤ỉ♥❣ ♥❣ø♥❣ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ tỉ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ✣✐♥❤ ❚❤à ❍✉②➲♥ ✷ ▼ð ✤➛✉ ▲❡♦♥❛r❞♦ P✐s❛♥♦ ❇♦❣♦❧❧♦ ✭❦❤♦↔♥❣ ✶✶✼✵ ✕ ỏ ữủ t ợ t r Ps t ữợ t ởt t ữớ ị ổ ữủ ởt sè ♥❣÷í✐ ①❡♠ ❧➔ ✧♥❤➔ t♦→♥ ❤å❝ t➔✐ ❜❛ ♥❤➜t t❤í✐ ❚r✉♥❣ ❈ê✧✳ ❋✐❜♦♥❛❝❝✐ ♥ê✐ t✐➳♥❣ tr♦♥❣ t❤➳ ❣✐ỵ✐ ❤✐➺♥ õ ổ tr ỵ số ❘➟♣ ð ❝❤➙✉ ❹✉✱ ✈➔ ✤➦❝ ❜✐➺t ❧➔ ❞➣② sè ❤✐➺♥ ✤↕✐ ♠❛♥❣ t➯♥ æ♥❣✱ ❞➣② ❋✐❜♦♥❛❝❝✐ tr♦♥❣ ❝✉è♥ s→❝❤ ▲✐❜❡r ❆❜❛❝✐✳ ❉➣② sè ❋✐❜♦♥❛❝❝✐ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ✈➫ ✤➭♣ ❝õ❛ ❦❤♦ t➔♥❣ ❚♦→♥ ❤å❝✳ ❉➣② ❋✐❜♦♥❛❝❝✐ ①✉➜t ❤✐➺♥ ✈➔ ❜✐➳♥ ❤â❛ ✈ỉ t➟♥ tr♦♥❣ tü ♥❤✐➯♥✱ ✈ỵ✐ r➜t ♥❤✐➲✉ t➼♥❤ ❝❤➜t ✤➭♣ ✈➔ ù♥❣ ❞ö♥❣ q✉❛♥ trå♥❣✳ ✣➳♥ ♥❛② ❝â r➜t ♥❤✐➲✉ ♠ð rë♥❣ ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ♥❤÷ ❞➣② k ✲❋✐❜♦♥❛❝❝✐✳✳✳ ❍➛✉ ❤➳t ♥❤ú♥❣ t➼♥❤ ❝❤➜t tèt ❝õ❛ ♥❤ú♥❣ ❞➣② ♥➔② ✤➲✉ ①✉➜t ♣❤→t tø ❞➣② ❋✐❜♦♥❛❝❝✐✳ ▼ët tỗ t s s ợ ▲✉❝❛s✳ ❉➣② ♥➔② ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ ✤➦❝ ❜✐➺t tr♦♥❣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✳ ❍❛✐ ❞➣② ♥➔② ❧➔ ❝❤ó♥❣ ❝â ♠è✐ ❧✐➯♥ ❤➺ ❝❤➦t ❝❤➩ ✈ỵ✐ ♥❤❛✉✳ ❚r♦♥❣ tü ♥❤✐➯♥ ❝â ♥❤✐➲✉ ❤✐➺♥ t÷đ♥❣✱ sü ✈➟t ①✉➜t ❤✐➺♥ trị♥❣ ✈ỵ✐ ❞➣② sè ❋✐❜♦♥❛❝❝✐✳ ❍➛✉ ❤➳t ❝→❝ ❜ỉ♥❣ ❤♦❛ ❝â sè ❝→♥❤ ❤♦❛ ❧➔ ♠ët tr♦♥❣ ❝→❝ sè ✸✱ ✺✱ ✽✳ ❙è ♥❤→♥❤ tø ♠ët ❝➙② ❦❤✐ ✤✐ tø ❣è❝ ❧➯♥ ♥❣å♥ ❝ơ♥❣ t❤÷í♥❣ t✉➙♥ t❤❡♦ ❞➣② ❋✐❜♦♥❛❝❝✐ tứ rỗ ✺✱ ✽✱ ✶✸ ♥❤→♥❤✳ ◆❤ú♥❣ ❝❤✐➳❝ ❧→ tr➯♥ ♠ët ♥❤➔♥❤ ụ tữỡ ự ợ số r ✈➠♥ ♥➔② ❝❤ó♥❣ t❛ ✤✐ t➻♠ ❤✐➸✉ ❝→❝ ❜➔✐ t♦→♥ r✐➯♥❣✱ ❜➔✐ t♦→♥ tê♥❣ q✉→t ✈➲ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ tr♦♥❣ ✤â ❝→❝ ❤➺ ♥❣❤✐➺♠ ❧➔ ❝→❝ sè ❋✐❜♦♥❛❝❝✐✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ❧↕✐ sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ sè ❋✐❜♦♥❛❝❝✐ ✈➔ sè ▲✉❝❛s✱ ❣✐ỵ✐ t❤✐➺✉ ❤❛✐ ❜➔✐ t♦→♥ ✼✼✾ ✈➔ ✽✵✹ ✈➔ ❧í✐ ❣✐↔✐ ❝õ❛ ❤❛✐ ❜➔✐ t♦→♥ ♥➔②✳ ❈→❝ ❦➳t q✉↔ ✤➣ ❜✐➳t ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t t❤❡♦ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✸❪✳ ❈❤÷ì♥❣ ✷ t❛ t➟♣ tr✉♥❣ ✤✐ t➻♠ ❤✐➸✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✱ ❧í✐ ❣✐↔✐ ❜➔✐ t♦→♥ ✸ tr♦♥❣ ❦❤✐ m = 3, tø ✤â ✤÷❛ r❛ ❞ü ✤♦→♥ ❧í✐ ❣✐↔✐ ❝❤♦ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳ ❈ö t❤➸ tr♦♥❣ ♣❤➛♥ ✷✳✶ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ tê♥❣ q✉→t✳ P❤➛♥ ✷✳✷ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ tr♦♥❣ tr÷í♥❣ ❤đ♣ m = ❤♦➦❝ 4✳ P❤➛♥ ✷✳✸ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❝❤♦ tr÷í♥❣ ❤đ♣ tê♥❣ qt õ ỵ P ❤➳t ✷✳✽ ❧➔ ❝→❝ ❦➳t q✉↔ ①♦❛② q✉❛♥❤ ✈✐➺❝ ❝❤ù♥❣ ỵ t q ❜✐➳t ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ✈✐➳t t❤❡♦ t➔✐ ❧✐➺✉ ❬✹❪✳ ✹ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❉➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❉➣② sè ỵ (Fn)n N ữủ ổ tự tr ỗ F0 = 0, F1 = 1, Fn+1 = Fn + Fn−1 , (n ≥ 1), ð ✤➙② Fn ❧➔ sè ❤↕♥❣ t❤ù n ❝õ❛ ❞➣② sè ❋✐❜♦♥❛❝❝✐✳ ❈→❝ sè ✤➛✉ t✐➯♥ ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐✿ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ứ tự tr ỗ t❛ ❝â Fn+2 − Fn+1 − Fn = 0, ✈ỵ✐ ♠å✐ n ≥ ❉♦ ✤â t❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ x2 − x − = ❤❛② x2 = x + ữỡ tr ợ xn−1 t❛ ✤÷đ❝ xn+1 = xn + xn−1 ✭✶✳✶✮ ❘ã r➔♥❣ ♥➳✉ ϕ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ t❤➻ − ϕ ❝ô♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮✳ ❉♦ ✤â ϕn+1 = ϕn + ϕn−1 ✈➔ (1 − ϕ)n+1 = (1 − ϕ)n + (1 − ϕ)n−1 n ❱ỵ✐ ♠é✐ ❝➦♣ sè t❤ü❝ a, b✱ t❛ ✤➦t Fa,b (n) = aϕn + b(1 − ϕ) ✳ ❑❤✐ ✤â t➜t ❝↔ ❝→❝ ❤➔♠ ♥➔② t❤ä❛ tự tr ỗ ❈→❝ ❤➔♠ Fa,b (n) = aϕn + b(1 − ϕ)n ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ s✐♥❤✳ ❚r♦♥❣ ✣à♥❤ ♥❣❤➽❛ ❞➣② ❋✐❜♦♥❛❝❝✐✱ số ữủ ữợ tr ỗ sỷ ổ õ ❦❤➠♥✳ ▼➺♥❤ ✤➲ s❛✉ ✤➙② ❝❤♦ t❛ ❝ỉ♥❣ t❤ù❝ t÷í♥❣ ♠✐♥❤ ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ❇✐♥❡t✳ ❈ỉ♥❣ t❤ù❝ ❇✐♥❡t ✤÷đ❝ sû ❞ư♥❣ ❤ú✉ ❤✐➺✉ tr♦♥❣ ❝→❝ ❝❤ù♥❣ ♠✐♥❤ s❛✉ ♥➔②✳ ▼➺♥❤ ✤➲ ✶✳✶✳✸✳ ❉➣② sè ❋✐❜♦♥❛❝❝✐ ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ Fn = √ n 1+ − √ √ n 1− ❉➣② ▲✉❝❛s ❧➔ ♠ët ❞➣② sè ✤÷đ❝ ✤➦t t➯♥ ♥❤➡♠ ✈✐♥❤ ❞❛♥❤ ♥❤➔ t♦→♥ ❤å❝ ❋r❛♥☛❝♦✐s ➆❞♦✉❛r❞ ❆♥❛t♦❧❡ ▲✉❝❛s ✭✶✽✹✷✲✶✽✾✶✮✱ ♥❣÷í✐ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❞➣② sè ❋✐❜♦♥❛❝❝✐✱ ❞➣② sè ▲✉❝❛s ✈➔ ❝→❝ ❞➣② t÷ì♥❣ tü✳ ●✐è♥❣ ♥❤÷ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ♠é✐ sè tr♦♥❣ ❞➣② ▲✉❝❛s ❜➡♥❣ tê♥❣ ❝õ❛ ❤❛✐ số trữợ õ số ỗ tữỡ ỳ sè ▲✉❝❛s ❧✐➲♥ ♥❤❛✉ s➩ ❤ë✐ tư ✤➳♥ ❣✐ỵ✐ ❤↕♥ ❜➡♥❣ t➾ ❧➺ ✈➔♥❣✳ ❚✉② ✈➟② ❦❤→❝ ✈ỵ✐ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ❤❛✐ sè ✤➛✉ t✐➯♥ tr♦♥❣ ❞➣② ▲✉❝❛s ❧➔ L0 = ✈➔ L1 = ✭tr♦♥❣ ❞➣② ❋✐❜♦♥❛❝❝✐ ❧➔ ✵ ✈➔ ✶✮✳ ❈❤➼♥❤ ✈➻ t❤➳ ♠➔ ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ sè ▲✉❝❛s s➩ ❦❤→❝ ✈ỵ✐ sè ❋✐❜♦♥❛❝❝✐✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❈❤♦ r, s ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❦❤→❝ ❦❤æ♥❣✳ s ự ợ (r, s) ữủ ❧➔✿ u0 (r, s) = 0, u1 (r, s) = 1, un (r, s) = run−1 + sun−2 (n ≥ 2) ❚r♦♥❣ tr÷í♥❣ ❤đ♣ (r, s) = (1, 1) t❛ ❦➼ ❤✐➺✉ sè ❤↕♥❣ t❤ù n ❝õ❛ ❞➣② ❧➔ Ln ✈➔ ❣å✐ ♥❣➢♥ ❣å♥ ❧➔ ❞➣② ▲✉❝❛s✳ ❚÷ì♥❣ tü ♥❤÷ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❞➣② ▲✉❝❛s ✤÷đ❝ ❝❤♦ ❜ð✐ ❝ỉ♥❣ t❤ù❝ s❛✉✳ ợ số ữỡ n, t ❝â Ln = √ 1+ n − √ 1− n ❚ø ▼➺♥❤ ✤➲ ✶✳✶✳✸ ✈➔ ▼➺♥❤ t õ ỵ s ỵ t❛ ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❝→❝ sè ❤↕♥❣ tê♥❣ q✉→t s ỵ ợ số ữỡ n > m, t õ FnLm = Fn+m + Fn−m ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n t❛ ✤➦t F−n = (−1)n Fn ✈➔ Ln = (−1)n Ln ✶✳✷ ❇➔✐ t♦→♥ ✼✼✾ ◆➠♠ ✶✾✾✺✱ t↕♣ ❝❤➼ ✏❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧②✑ sè ✸✸✳✶ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❇✳✼✼✾ ❝õ❛ ❆♥❞r❡✇ ❈✉s✉♠❛♥♦✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✤â ❧➔✿ ❚➻♠ ❝→❝ sè ♥❣✉②➯♥ a, b, c ✈➔ d t❤ä❛ ♠➣♥ < a < b < c < d s ỗ t tự s ú ợ số ữỡ n Fn = Fna + 6Fn−b + Fn−c + Fn−d ✭✶✳✷✮ ✣➣ ❝â ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤→❝ ♥❤❛✉ ❣û✐ ❧í✐ ❣✐↔✐ ✤➳♥ t↕♣ ❝❤➼ ✏❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧②✑✱ ❤➛✉ ❤➳t ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ❝❤➾ ❣û✐ ✤➳♥ ❧í✐ ❣✐↔✐ a = 2, b = 5, c = 6, d = ✈➔ ❦❤➥♥❣ ✤à♥❤ r➡♥❣ ✈✐➺❝ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ Fn = Fn−2 + 6Fn−5 + Fn−6 + Fn−8 ✭✶✳✸✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤ù♥❣ ♠✐♥❤ q✉② ♥↕♣ t❤❡♦ n ❧➔ ✤ì♥ ❣✐↔♥✳ ❈❤➾ ❝â ❇r✉❝❦✲ ♠❛♥ ✈➔ ❋✐❣❣❤✐♦♥ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ❝ö t❤➸ ✈➔ ❝❤➾ r❛ ❝→❝❤ t➻♠ a, b, c, d✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ t✐➳♣ ❝➟♥ ✈➔ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ❞÷í♥❣ ♥❤÷ ❦❤ỉ♥❣ ❝â t➼♥❤ ❦❤→✐ q✉→t✳ ❚❛ ❝â t❤➸ ❝â ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝ ✭✶✳✸✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② t n ữ s ợ n = t ữỡ tr tữỡ ữỡ ợ F8 = F6 + 6F3 + F2 + F0 ✣➥♥❣ t❤ù❝ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣ ✈➻ F8 = 21, F6 = 8, F3 = 2, F2 = 1, F0 = ●✐↔ sû ✤➥♥❣ t❤ù❝ ✤ó♥❣ ✈ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ≤ k ≤ n ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✭✶✳✸✮ ✤ó♥❣ ✈ỵ✐ k = n + ❚❤❡♦ ✣à♥❤ ♥❣❤➽❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❣✐↔ t❤✐➳t q✉② ✼ ♥↕♣ t❛ ❝â Fn+1 = Fn + Fn−1 = (Fn−2 + 6Fn−5 + Fn−6 + Fn−8 ) + F(n−1)−2 + 6F(n−1)−5 + F(n−1)−6 + F(n−1)−8 = (Fn−2 + Fn−3 ) + (Fn−5 + Fn−6 ) + (Fn−6 + Fn−7 ) + (Fn−8 + Fn−9 ) = Fn−1 + 6Fn−4 + Fn−5 + Fn−7 ❱➻ ✈➟② t❛ ❝â Fn+1 = F(n+1)−2 + 6F(n+1)−5 + F(n+1)−6 + F(n+1)−8 ✶✳✸ ❇➔✐ t♦→♥ ✽✵✹ ◆ë✐ ❞✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✽✵✹ ❧➔✿ ❍➣② t➻♠ t➜t ❝↔ ❝→❝ sè ♥❣✉②➯♥ a, b, c ✈➔ d ✭✈ỵ✐ < a < b < c < d s ỗ t tự s ú ợ số ữỡ n Fn = Fn−a + 9342Fn−b + Fn−c + Fn−d ✭✶✳✹✮ ◆❣❛② s❛✉ ✤â✱ ♥➠♠ ✶✾✾✼✱ ▲✳❆✳●✳ ❉❡rs❡❧ ✤➣ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ ❝õ❛ ❜➔✐ t♦→♥ ✽✵✹ tr♦♥❣ sè ✸✺✳✶ ✭✶✾✾✼✮ ❝õ❛ t↕♣ ❝❤➼ ❚❤❡ ❋✐❜♦♥❛❝❝✐ ◗✉❛rt❡r❧②✳ ▲í✐ ❣✐↔✐ ❝ư t❤➸ ♥❤÷ s❛✉✳ ❚ø ♥❤➟♥ ①➨t 9342 = 9349 − = L19 − L4 , ð ✤➙② Lk ❧➔ số s tự ỷ ỗ t tự ❣✐ú❛ ❝→❝ sè ❋✐❜♦♥❛❝❝✐ ✈➔ sè ▲✉❝❛s t❛ ❝â Fm+19 − Fm−19 = Fm L19 , Fm+4 + Fm−4 = Fm L4 ❚rø ✈➳ ✈ỵ✐ ✈➳ ❝õ❛ ✷ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ♥❤➟♥ ✤÷đ❝ Fm+19 − Fm−19 − Fm+4 − Fm−4 = Fm (L19 − L4 ) ✣➦t n = m + 19✱ t❛ ♥❤➟♥ ✤÷đ❝ ✤➥♥❣ t❤ù❝ s❛✉ Fn = Fn−15 + 9342Fn−19 + Fn−23 + Fn♦ ❇ê ✤➲ ✷✳✻✳✷ ✭✐✈✮ ❝â ♠ët sè ♥❣✉②➯♥ ❧ỵ♥ ♥❤➜t p✱ ✈ỵ✐ ≤ p ≤ k − s❛♦ ❝❤♦ {x(k), x (k − 1) , , x (k − (p − 1))} = {b − 1, b − 2, , b − p} ✭✷✳✶✸✮ ❚❛ t❤➜②✱ ♥➳✉ k − (p − 1) > 3, t❤➻ tø t➼♥❤ ❝❤➜t ❧ỵ♥ ♥❤➜t ❝õ❛ p✱ t❛ ❝â x (k − p) ≤ b − p − ❚÷ì♥❣ tü✱ ♥➳✉ k − (p − 1) = t❤➻ x(1) ≤ b − p − ❚❤➟t ✈➟②✱ ❝❤♦ k − (p − 1) = ✈➔ ❣✐↔ sû ♥❣÷đ❝ ❧↕✐ r➡♥❣ x(1) = b − p − 2✳ ❚❤❛② ✭✷✳✶✸✮ ✈➔♦ ✭✷✳✶✵✮ t❛ ✤÷đ❝ Fb−2 = Fb−1 − Fb−2 + Fb−3 − Fb−4 , ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❇ê ✤➲ ✷✳✸✳✹✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝➜✉ tró❝ ❝õ❛ S1 ❜➡♥❣ ❝→❝❤ ①➨t ❜❛ tr÷í♥❣ ❤đ♣ p tữợ k (p 1)✳ ❇ê ✤➲ ✷✳✻✳✸✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮ ❧➔ ❝è ✤à♥❤✳ ❑❤✐ ✤â ❝→❝ ♣❤→t ❜✐➸✉ s❛✉ ❧➔ ✤ó♥❣ ✭✐✮ ◆➳✉ p ❧➔ ❝❤➤♥ ✈➔ k − (p − 1) = t❤➻ {x(1), x(3), x(4), , x(k)} = {b − o − 3, b − o − 1, b − o, , b − 1} ✭✷✳✶✹✮ ✭✐✐✮ ◆➳✉ p ❧➔ ❝❤➤♥ ✈➔ k − (p − 1) > t ợ ởt số ữỡ j ✤â✱ {x(1), x(3), x(4), , x(j), x (j + 1) , x (j + 2) , , x(k)} = {b − o − − o , b − o − − o , b − o − − o , , b − o − 4} ✭✷✳✶✺✮ ∪ {b − o − 1, b − o, , b − 1} ✭✐✐✐✮ p ❦❤æ♥❣ t❤➸ ❧➔ sè ❧➫✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â Fx(1) = Fb−2 − F−x(k) + F−x(k−1) + + F−x(k−(p−1)) , ✭ ❜ð✐ ✭✷✳✶✵✮✮ = Fb−2 − (Fb−1 − Fb−2 ) + (Fb−3 − Fb−4 ) + + Fb−(p−1) − Fb−p = Fb−2 − Fb−3 − Fb−5 − Fb−7 − − Fb−(p+1) = Fb−(p+2) ✭t❤❡♦ ❇ê ✤➲ ✷✳✸✳✹✮, ✸✷ ð ✤➙② ❞➜✉ ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ ❧➔ ❞♦ ✭✷✳✶✸✮✳ ✣➦t o = p + t❤➻ t❛ ❝â ✭✷✳✶✹✮✳ ✭✐✐✮ ❚÷ì♥❣ tü ♥❤÷ tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ♣❤➛♥ ✭✐✮ t❛ ❝â Fx(1) + F−x(3) + F−x(4) + + F−x(k−p) = Fb−2 − F−x(k) + F−x(k−1) + + F−x(k−(p−1)) = Fb−2 − (Fb−1 − Fb−2 ) + (Fb−3 − Fb−4 ) + + Fb−(p−1) − Fb−p = Fb−2 − Fb−3 − Fb−5 − Fb−7 − − Fb−(p+1) = Fb(p+2) rữợ t t r r x(j) ≤ b − (p + 3) ✈ỵ✐ ≤ j ≤ k − p✳ ❉♦ t➼♥❤ tè✐ ✤↕✐ ❝õ❛ p tr♦♥❣ ✭✷✳✶✸✮ t❛ ❝â x(j) ≤ b − (p + 2) ✈ỵ✐ ≤ j ≤ k − p✳ ❍ì♥ ♥ú❛✱ ♥➳✉ x (k − p) = b − (p + 2) ú t ỗ t tự s ổ ú ỗ t tự s➩ ❦❤æ♥❣ ①↔② r❛✳ ❱➻ t❤➳ x(j) ≤ b − (p + 3)✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤➾ r❛ r➡♥❣ x(1) ≤ b − (p + 3)✳ ◆➳✉ x(1) = b − (p + 2) t❤➻ ✭✷✳✶✻✮ s➩ ❦❤ỉ♥❣ ✤ó♥❣✳ ▼➦t ❦❤→❝ ♥➳✉ x(1) ≥ b − (p + 1) t❤➻ ❇ê ✤➲ ✷✳✸✳✻ ❝❤♦ ✭✷✳✶✻✮ t❛ ❝â Fx(1) + i∈J Fi = i∈K≤b−(p+2) Fi < Fb−(p+1) ≤Fx(1) , ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❱➟② x(1) ≤ b − (p + 3)✳ ❱➻ x(1) ≤ b − (p + 3) ✈ỵ✐ i ỵ ✈ỵ✐ b − (p + 2) t❤❛② t❤➳ ❝❤♦ b✱ s✉② r❛ ✈ỵ✐ ♠ët sè ❝❤➤♥ q t❛ ❝â x(1) = (b − p − 2) − q, x(3) = (b − p − 2) − q + 1, x(4) = (b − p − 2) − q + 3, ··· , x(k − p) = (b − p − 2) − ❳→❝ ✤✐♥❤ o ✈➔ o tø ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ (b − p − 2) − q = b − o − − o ✈➔ b − o − = b − q − 3✳ ❱➻ b✱ q ✱ ✈➔ p ❝❤➤♥ ♥❣❛② ❝↔ ❦❤✐ o ✈➔ o ❧➔ ❧➫✳ ❑➳t ❤ñ♣ ♥❤ú♥❣ ❦➳t q✉↔ ♥➔② ợ t õ ỗ t tự ... ❞➣② sè ▲✉❝❛s ✈➔ ❝→❝ ❞➣② t÷ì♥❣ tü✳ ●✐è♥❣ ♥❤÷ ❞➣② ❋✐❜♦♥❛❝❝✐✱ ♠é✐ sè tr♦♥❣ ❞➣② ▲✉❝❛s ❜➡♥❣ tờ số trữợ õ số ỗ t❤÷ì♥❣ ❣✐ú❛ ❤❛✐ sè ▲✉❝❛s ❧✐➲♥ ♥❤❛✉ s➩ ❤ë✐ tư ✤➳♥ ❣✐ỵ✐ ❤↕♥ ❜➡♥❣ t➾ ❧➺ ✈➔♥❣✳ ❚✉② ✈➟② ❦❤→❝... n, t❛ ❝â Ln = √ 1+ n − √ 1− n ❚ø ▼➺♥❤ ✤➲ t õ ỵ s ỵ t ố ỳ số ❤↕♥❣ tê♥❣ q✉→t ❝õ❛ ❞➣② ❋✐❜♦♥❛❝❝✐ ✈➔ ❞➣② ▲✉❝❛s✳ ✻ ỵ ợ số ữỡ n > m, t❛ ❝â FnLm = Fn+m + Fn−m ❱ỵ✐ ♠é✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ n t❛... s❛✉✳ ❚ø ♥❤➟♥ ①➨t 9342 = 9349 − = L19 − L4 , ð ✤➙② Lk ❧➔ sè ▲✉❝❛s t❤ù ❦✳ ❙û ❞ö♥❣ ❝→❝ ỗ t tự ỳ số số s t❛ ❝â Fm+19 − Fm−19 = Fm L19 , Fm+4 + Fm−4 = Fm L4 ❚rø ✈➳ ✈ỵ✐ ✈➳ ❝õ❛ ✷ ✤➥♥❣ t❤ù❝ tr➯♥ t❛