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(·Ó¤ù¨ú¦Û¡uThe MacTutor History of Mathematics Achieve¡vhttp://www-gap.dcs.st-and.ac.uk/~history/ )
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©Ò¿×¶Oªi®³«´¼Æ¦C (Fibonacci Sequence) ²ºÙ Fn¡A§Y 1, 1, 2, 3, 5, 8, 13, 21,... (OEIS A000045)
·í¤¤ F1 = F2 = 1¡AFn+2 = Fn+1 + Fn (¨ä¤¤ n = 1,2,3,...)
§Ṳ́]¥i©w¸q F0 = 0 ¡C
·í¤¤ªº¼Æ«K¬O¶Oªi®³«´¼Æ (Fibonacci Number)¡C
¥¦¬O 13¥@¬ö·N¤j§Q¼Æ¾Ç®a¶Oªi®³«´ (Leonardo Pisano Fibonacci ¬ù1170-¬ù1250) ¬ã¨s¤@¦³½ìªº°ÝÃD¦Ó²£¥Í¡A¬G¥H¤§¬°¦W¡C
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³]¨C¤@¹ï¨ß¤l¨C¤ë·|²£¤U¤@¹ï¤p¨ß¡A¤p¨ß¤@Ó¤ë«á·|ªø¤jÅܦ¨¤j¨ß¡C¦pªG¨S¦³¦º¤`¡A°Ý¥Ñ¤@¹ï¤j¨ß¶}©l¡A¤@¦~«á¦³¦h¤Ö¹ï¤j¨ß©O¡H
§Ṳ́£§«³]¶}©l®É¤j¨ßªº¹ï¼Æ¬° F1 ¡A¤@Ó¤ë«áªº¤j¨ßªº¹ï¼Æ¬° F2¡A¨âÓ¤ë«áªº¤j¨ßªº¹ï¼Æ¬° F3¡A¤@¯ë¦a n Ó¤ë«áªº¤j¨ßªº¹ï¼Æ¬° Fn+1 ¡C
¥ÑÃD¥iª¾ F1 = 1¡A¤@Ó¤ë«á¡A¨e̤F¥Í¤F¤@¹ï¤p¨ß¡A¦ý¤j¨ßªº¹ï¼Æ¤´¬° 1¡A§Y F2 = 1¡C¦A¹L¤F¤@Ó¤ë¡A¤p¨ßªø¤j¤F¡A¤j¨ß¤S¥Í¤F¤@¹ï¤p¨ß¡A¤j¨ßªº¹ï¼Æ¦¨¤F 2 ¡A§Y F3 = 2 ¡C¦p¦¹¤U°_¡A§Ú̦³ F4 = 3 ¡B F5 = 5 ¡B F6 = 8 ....
¤@¯ëªº¡A²Ä n+1 Ó¤ë¥H«áªº¤j¨ß¹ï¼Æ¬O¥]§t¤F¨âÓ³¡¥÷¡G¤@¬O쥻²Ä n Ó¤ë¥H«áªº¤j¨ß¹ï¼Æ¡A¤G¤WӤ몺¤p¨ß¹ï¼Æ¡A¦]¨e̦¨ªø¤F¡A¦¨¤F¤j¨ß¡A³o¹ê¬O²Ä n-1 Ó¤ë¥H«áªº¤j¨ß¹ï¼Æ¡C¬G¦³ Fn+2 = Fn+1 + Fn ¡C
³o¼Æ¦C Fn «K¬O§Ú̲{¦b©Ò¨¥¤Îªº¶Oªi®³«´¼Æ¦C¡C¦^¨ì°ÝÃD¡A¤@¦~¥H«á¡A§Y 12 Ó¤ë¥H«á¡A§Y F13 = 233 «K¬O§ÚÌAªºµª®×¤F¡C
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¶Oªi®³«´¼Æ¦Cªº¬Û¾F¨â¶µ¤§¤ñ¬O·|ÁͦV¤@Ó·¥Èªº¡A¦Ó³oÓ·¥È¥¿¬O§Ú̸g±`»¡ªº¶Àª÷¤ñ (Golden Ratio)¡A§Y f = 1.618033988749894848204586834365 ...... (OEIS A001622)¡C³o±`¼Æ¦P¬O¥ç¬O¤èµ{ x2 - x - 1 = 0 ªº¤@Ó¥¿¹ê®Ú¡C
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¨ä¹ê¶Oªi®³«´¼Æ¦³«Ü¦h¯S§Oªº¦a¤è¡G
F1
+ F2 + F3 +...+ Fn = Fn+2 -1
F2
+ F4 + F6 +...+ F2n = F2n+1 -1
F1
+ F3 + F5 +...+ F2n-1 = F2n
F12
+ F22 + F32 +...+ Fn2
= Fn Fn+1
Fn+12
= FnFn+2 + (-1)n
F2n
= Fn+12 - Fn-12
F3n
= Fn+13 + Fn3 - Fn-13
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·í¤¤¥ç¦³¤@¨ÇµÛ¦WªºùÚµ¥¦¡©M¶Oªi®³«´¼Æ¬ÛÃö¡G
Fn2
- Fn+rFn-r = (-1)n-rFr2
(¥d¶ðÄõùÚµ¥¦¡ Catalan's Identity)
FmFn+1
- Fnm+1 = (-1)nFn+m (}¶ø¥d¥§ùÚµ¥¦¡ d'Ocagne's
Identity)
Fn4
- Fn-2Fn-1Fn+1Fn+2 = 1 (®æªL - ¤ÁÂÄùùÚµ¥¦¡
Gelin - Cesaro's Identity)
Fn-1Fn+1
- Fn2 = (-1)n (¥d¦è¥§ùÚµ¥¦¡ Cassini's Identity)
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§ÚÌ¥ç¥i¥H¤U¦¡p¥X¶Oªi®³«´¼ÆªºÈ¡G
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¶Oªi®³«´¼Æ¦Cªº¥Í¦¨¨ç¼Æ (Generating Function) ¬°¡G
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§ÚÌ¥ç¥i±q©¬´µ¥d¤T¨¤§Î (Pascal's Triangle) ¤¤§ä¨ì¶Oªi®³«´¼Æ¦C¡G
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¶Oªi®³«´¯À¼Æ
¦Ó§Ú̦³¿³½ì¬O·í¤¤¥X²{ªº¯À¼Æ¡GF3=2 ¡B F4=3 ¡B F5=5 ¡B F7=13 ¡B F11=89 ¡B F13=233 ¡B F17=1597 ¡BF23=28657 ¡B F29=514229 ¡B ... §Ú̺ٳo¨Ç¯À¼Æ¬°¶Oªi®³«´¯À¼Æ (Fibonacci Prime)¡C (OEIS A005478 ¤Î A001605)
§Ṳ́£Ãøµo²{°£¤F F4 = 3 ¥H¥~¡A¨ä¾lªº²Ä p Ó¶Oªi®³«´¼Æ ( p ¬O¯À¼Æ) Fp ¤]¬O¯À¼Æ¡Cì¨Ó©Ò¦³¶Oªi®³«´¯À¼Æ¡A°£¤F F4 = 3 ¥H¥~¡A³£¦³¤@¯À§Ç¼Æ (Prime Order)¡C¦ý¤Ï¤§«h¥¼¥²¦¨¥ß¡A¦p F19 = 4181 = 37 * 113¡B F31 = 1346269 = 557 * 2417 ¡G§Y¤£¬O©Ò¦³¦³¯À¦¸¼Æªº¶Oªi®³«´¼Æ¤]¬O¯À¼Æ¡C§ÚÌ ¤wª¾ Fn ¬O Fnm ªº¦]¤l¡C¬O¬G§ÚÌ°Q½×ªº¶Oªi®³«´¯À¼Æ¥þ¬O¥Ñ Fp ©Î°O¦¨ F(p) ¤¤§ä¥X¨Óªº¡A¨ä¤¤ p ¬O¯À¼Æ¡C¦ý¬O§_¦³µL¦hÓ¶Oªi®³«´¯À¼Æ¤]¬O¥¼ª¾¤§¨Æ¡A¦ý¼Æ¾Ç®a̶ɦV»{¬°³o¦s¦³µL¦hÓ¶Oªi®³«´¯À¼Æ¡C
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Caldwell, C. K. "The Top Twenty: Fibonacci Number." http://primes.utm.edu/top20/page.php?id=39.
Gardner, M. Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. New York: Knopf, 1979.
Weisstein, E. W. "Fibonacci Number." From MathWorld. http://mathworld.wolfram.com/FibonacciNumber.html.
Weisstein, E. W. "Golden Ratio." From MathWorld. http://mathworld.wolfram.com/GoldenRatio.html.
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