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1 , 2 August 28 (Fri), 2016 August 28 (Fri), / 64
2 Outline August 28 (Fri), / 64
3 fibonacci Lucas 2 August 28 (Fri), / 64
4 Dynamic Programming R.Bellman Bellman Continuum ( ) Eye of the Hurricane, an autobiography (WikipediA ) August 28 (Fri), / 64
5 August 28 (Fri), / 64
6 ,,,,,,,, (The Da Vinci Code by Dan Brown 2003 )!! (Wikipedia) 1,1,2,3, 5, 8, 13, 21 34, 55, 89, n F n August 28 (Fri), / 64
7 4 3 August 28 (Fri), / 64
8 4 3 4 ( ± 1 ) 2 ( 1 = 2 r 1 r 2 r 3 r 4 r r r ) r4 2 3 (r 3 ) 4 r = r 4 1 r = 1 r r 2 1 = F n 1 + F n 2 r n r 2 r 1 August 28 (Fri), / 64
9 August 28 (Fri), / 64
10 Fibonacci The Man August 28 (Fri), / 64
11 Fibonacci Statue in Pisa August 28 (Fri), / 64
12 Who is the Man? August 28 (Fri), / 64
13 Fibonacci( ): Leonard of Pisa Èdouard Lucas( ):Fibonacci sequence and the test for Mersenne primes The Fibonacci Association Official Website 1963 The Fibonacci Quarterly, Internationa Conferences from 1984 biyearly : OEIS(On-line Encyclopedia of Integer Sequences), founded by in 1964 by N.J.A.Sloane August 28 (Fri), / 64
14 L.E.Sigler translation by Google Scholar Leonardo Pisano (Fibonacci): The Book of Squares(2013) Fibonacci s Liber abaci: a translation into modern English of Leonardo Pisano s Book of calculation(2002): August 28 (Fri), / 64
15 August 28 (Fri), / 64
16 Édouard Lucas August 28 (Fri), / 64
17 Édouard Lucas August 28 (Fri), / 64
18 R.Knott Homepage: August 28 (Fri), / 64
19 2 (Kiefer) P.Whittle Optimization over Times: Dynamic Programming and Stochastic Control, Wiley ,, Integer Partitions by G.E.Andrews. Pattern Identities, A.D.Healy, math192.pdf 2001, (k, l)-sequence August 28 (Fri), / 64
20 3 Kalman Filter; ( ) Riccati J.Donoghue; The Kalman Filter for Complex Fibonacci Systems, ISRN Signal Processing vol.2012, Article ID , 5 page. Ladder( ) network : Morgan-Voyce; Lader network analysis using Fibonacci numbers, Proc.IRE, IRE Trans, on Circuit Theory, Sep,1959, pp August 28 (Fri), / 64
21 Fibonaci n ; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 F n ; 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 Lucas n ; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 L n ; 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 (n = 0, 1) n ; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, x n ; 100, 1, 101, 102, 203, 305, 508, 813, 1321, 2134, 813 = F F 7, 1321 = F F 8, 3455 = F F 10, August 28 (Fri), / 64
22 x 2 x 2 x 1 = 1 + x + 2x 2 + 3x 3 + 5x 4 + 8x = = = August 28 (Fri), / 64
23 Lucas sequence: x n = p x n 1 q x n 2 (i) Fibonacci number p = 1, q = 1 x n = x n 1 + x n 2 x 0 = 0, x 1 = 1, (ii) Lucas number p = 1, q = 1 x n = x n 1 + x n 2 x 0 = 2, x 1 = 1, (iii) Pell number p = 2, q = 1 x n = 2x n 1 + x n 2 x 0 = 0, x 1 = 1, x 2 = 2, x 3 = 5, x 4 = 12, (vi) Mersenne number p = 3, q = 2 x n = 3x n 1 2x n 2 x 0 = 0, x 1 = 1, x 2 = 3, x 3 = 7, x 4 = 15, M 2 n 1 August 28 (Fri), / 64
24 F 6 = 2 3, F 8 = 3 7, F 9 = 2 17, F 10 = 5 11, F 12 = , F 14 = 13 29, F 15 = , F 18 = , F 20 = , F 21 = , F 24 = , F 28 = , F 30 = , F 35 = , F 36 = , F 42 = , F 70 = , (J.H.Halton; FQ 1966) August 28 (Fri), / 64
25 1 Fn 2 F n 1 F n+1 = ( 1) n 1 2 GCM(F m, F n ) = F GCM(m,n) 3 August 28 (Fri), / 64
26 August 28 (Fri), / 64
27 Fibonacci Polynomial Hoggatt,V.Jr.,Long,C.T.(FQ, 1974) Definition 1 f n+2 (x) = x f n+1 (x) + f n (x); f 0 (x) = 0, f 1 (x) = 1. u n+2 (x, y) = x u n+1 (x, y) + y u n (x, y); u 0 (x, y) = 0, u 1 (x, y) = 1. August 28 (Fri), / 64
28 Fibonacci Polynomial 2 n u n (x, y) F n = u n (1, 1) x 1 3 x 2 + y 2 4 x 3 + 2xy 3 5 x 4 + 3x 2 y + y x 5 + 4x 3 y + 3xy x 6 + 5x 4 y + 6x 2 y 2 + y 3 13 August 28 (Fri), / 64
29 Hoggatt and Long (FQ 1974) Corollay 10 (page 118) For n 2, n even, and for n 2, n odd, u n (x, y) = x Π (n 2)/2 k=1 u n (x, y) = x Π (n 1)/2 k=1 ( x 2 + 4y cos 2 kπ n ( x 2 + 4y cos 2 kπ n ) ) August 28 (Fri), / 64
30 Real number sequences; F n = 1 {ϕ n ( 1/ϕ) n } 5 ( = Π [(n 1)/2] k= cos 2 kπ ) n L n = ϕ n + ( 1/ϕ) n n = 1, 2, n = 1, 2, where Goledn Raio: ϕ = August 28 (Fri), / 64
31 Continuous Fibonacci sequence August 28 (Fri), / 64
32 Extended Fibonacci numbers cos(nπ) = ( 1) n (n = 0, ±1, ±2, ), x F x = 1 { ϕ x 1 } 5 ϕ x cos(πx) L x = F x 1 + F x+1 J.Donoghue ISRN Signal Processing 2012 ( 1) x = e iπx = cos(πx) + i sin(πx), i = 1 August 28 (Fri), / 64
33 Complex Fibonacci numbers and Lucas REFERENCE: Horadam,A.F.,Shannon,A.G.:Finbonacci and Lucas Curves, FQ, vol , Good,I.J.:Complex Fibonacci and Lucas Numbers, Continued Fractions, and the Square Root of the Golden Ratio(Condensed Version), J. Opr. Res Soc. 1992(43) , Garnier,N. and O.Pamarè(FQ 2008): Fibonacci numbers and trigonometric identities, intrigue(!) How could we connect cos 2kπ n cos 2kπ n+1? and August 28 (Fri), / 64
34 Complex Fibonacci numbers and Lucas REFERENCE2: DISMAY( ): Fibonacci numbers are linked with the arithmetic of Q( 5) and not with that of Q(exp(2iπ/n)). Horadam,A.F.: A Generalized Fibonacci Sequence, Amer Math Month 1961 (vol. 68(5) ), Complex Fibonacci Numbers and Finobacci Quaternions, Amer Math Month 1963 (vol.70(3) ). August 28 (Fri), / 64
35 Dymanic Programming by R.Bellman WikipediA : (1) optimal substructure (2) principle of optimality August 28 (Fri), / 64
36 2 2 c x n 1 x n 3 c x n 1 x n 3 x n 5 x 2 = x n x n 2 x n 4 2 (xn 2 + xn 1 2 ) (x n x n 3 2 ) c x n c = x n + x n 1 c x n = x n 1 [ min (x 2 x n,x n 1,x n 2, n + xn 1) 2 + (xn xn 3) 2 + ] August 28 (Fri), / 64
37 ( ) + ( ) = 27 August 28 (Fri), / 64
38 ( ) + ( ) = 15 27( ) August 28 (Fri), / 64
39 ( ) + ( ) + ( ) = 171 August 28 (Fri), / 64
40 ( ) + ( ) + ( ) = 108 August 28 (Fri), / 64
41 2 c 13 x (c x 1 ) x (x 1 x 2 ) x (x 2 x 3 ) ( ) + ( ) + ( ) = 104 August 28 (Fri), / 64
42 2 f 2 n + f 2 n 1 + f 2 n 2 + f 2 n f 2 1 = f n f n = 104 = 8 13 c = f n a = f n 1 f n 2 Backward f n f n 1 = f n 2 : f n [n ] [(n 1) a = f n 1 ] = f n 2 [(n 2) ], August 28 (Fri), / 64
43 2 Cassini ( 1) n = f n+1 f n 1 f 2 n ( ) n ( ) 1 1 fn+1 f = n 1 0 f n f n 1 ( ) a b t = a t + b c d c t + d ( ) August 28 (Fri), / 64
44 2 z n = f n /g n z n+1 = 2z n + 1 z n + 1 f n+1 = f n + (f n + g n ) g n+1 f n + g n Lucas L 2 n + L 2 n 1 + L 2 n 2 + L 2 n L 2 1 = L n L n+1 L 0 August 28 (Fri), / 64
45 2 c = F 5 = 5 F 5 F 4 F 5 F 4 = F F F F = 5 3 = c = F 8 = 13 F 5 F 4 F 8 F 7 = F F F F = 13 8 = August 28 (Fri), / 64
46 2 1 = x + (1 x) min {Ax 2 + B(1 x) 2 } = C 0 x 1 x = B A + B, 1 x = A A + B C = A B A + B A = 1/F n, B = 1/F n 1 { } x 2 (1 x)2 min + = 1 0 x 1 F n F n 1 F n+1 3 August 28 (Fri), / 64
47 2 min 0 x,y,x+y=1 { x 2 + y 2 } = 1 F n F n 1 F n+1 {T n : T n = T n 1 + T n 2 + T n 3 } { x 2 min + y x,y,z,x+y+z=1 T n T n 1 z2 T n 2 } = 1 T n+1 { min Ax 2 + By 2 + Cz 2} = 0 x,y,z,x+y+z=1 = ABC AB + BC + CA 1 1/A + 1/B + 1/C August 28 (Fri), / 64
48 2 2 ( ), 2013 ( ), ( ), 25 RIMS ( ), ( ), (), 25 RIMS August 28 (Fri), / 64
49 (run): (avoiding sequence) Havil( Gamma,Princeton Univ) p.119 (Doctrine) ( (1717) 74) n k P(10, 3) = = T (13) = P(21, 4) = = 1 T (25) 2 21 P(n, k) = {n k } T (n) August 28 (Fri), / 64
50 2 2 2 n {0, 1} n F (n + 2) F (n) n August 28 (Fri), / 64
51 1 n = 3 {0, 1} 2 3 = 8 11 F (3 + 2)/2 3 = F (5)/8 = 5/8 x 1 x 2 x 3 xi x i August 28 (Fri), / 64
52 2 n = 4 F (4 + 2)/2 4 = F (6)/16 = 8/16 x 1 x 2 x 3 x 4 xi x i August 28 (Fri), / 64
53 , 11 2 n X i Binom ( 1, 1 2), i = 1, 2,, n P ( n 1 ) X i X i+1 = 0 = i=1 F (n) n-th F (n + 2) 2 n n 1 i=1 (X i X i+1 ) = 0 August 28 (Fri), / 64
54 {a n, b n } n=1,2, { an+1 = a n + b n b n+1 = a n a n + b n = F (n + 2) ( ) n 1 1 = 1 0 ( F (n + 1) F (n) F (n) F (n 1) ) n = 1, 2, August 28 (Fri), / 64
55 n (1) n 0 a n (2) n 1 b n a n + b n n + 1 a n 0 1 b n 0 { an+1 = a n + b n b n+1 = a n August 28 (Fri), / 64
56 ( ) ( ) ( ) an an = b n+1 ( 1 0 ) b n n ( ) 1 1 a1 = ( 1 0 b 1 F (n + 1) F (n) = ( F (n) ) F (n 1) F (n + 2) = F (n + 1) ) ( 1 1 ) n a n + b n = F (n + 1) + F (n) = F (n + 2) (QED) 4 August 28 (Fri), / 64
57 (1) n 0 a n = F (n + 1). (2) n 1 b n = F (n). August 28 (Fri), / 64
58 C = min{ax 2 + B(1 x) 2 ; 0 x 1} = i.e. x = B A + B, 1 x = 1 A + 1 B = 1 C A A + B 1 1/A + 1/B, A, B C x 1, , 3/ , 5/ , 8/ August 28 (Fri), / 64
59 Havil ( ) P(n, k) = P(n 1, k) + {1 P(n k 1, k)}/2 k+1 P(0, k) = P(1, k) = = P(k 1, k) = 0 P(n, k) = 1 F k (n + k)/2 n F k k k = 2, 3, August 28 (Fri), / 64
60 T 0 = T 1 = 0, T 2 = 1, T n+3 = T n + T n+1 + T n+2, (n 0) n T n August 28 (Fri), / 64
61 (1) (2) (1) (2) (3) T n = + + August 28 (Fri), / 64
62 StringReplace StringCount MatrixPower[mat, n] n a n (1, 1) a n (1, 2) a n (1, 3) = a (2, 1) a n (2, 2) a n (2, 3) a n (3, 1) a n (3, 2) a n (3, 3) August 28 (Fri), / 64
63 n T n+2 T n+1 + T n T n+1 = T n+1 T n + T n 1 T n T n T n 1 + T n 2 T n 1 T n+3 = T n+2 + T n+1 + T n T n T n+2 = T n+2 T n+2 = T n+1 = T n+1 T n T n+2 T n+1 T n August 28 (Fri), / 64
64 Theorem n X i Binom ( 1, 1 2), i = 1, 2,, n P ( n 2 ) X i X i+1 X i+2 = 0 = i=1 T (n) n-th T (n + 3) 2 n n August 28 (Fri), / 64
Talk/7Akita-cont.tex Dated: 7/Feb/ Fibonacci Quartery L n = i n T n i/).) F n = i n U n i/).6).),.6) n = 7, F 7 = F n = cos π ) cos π 7 7 ) F = 8 [n )
Talk/7Akita-cont.tex Dated: 7/Feb/ wikipedia Pafnuty Lvovich Chebyshev 8 6-89 8 6 Chebychev Chebyshov Tchebycheff Tschebyscheff https://ja.wikipedia.org/wiki/ wikipediaa) 96 cos π 7 cos π 7 cos π 7 = x
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