離散数理工学 第 2回 数え上げの基礎:漸化式の立て方
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- さゆり しばもと
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1 :48 ( ) (2) / 44
2 ( ) 1 (10/7) ( ) (10/14) 2 (10/21) 3 ( ) (10/28) 4 ( ) (11/4) 5 (11/11) 6 (11/18) 7 (11/25) ( ) (2) / 44
3 ( ) 8 (12/2) 9 (12/9) (12/16) 10 ( ) (1/6) 11 ( ) (1/13) 12 ( ) (1/20) ( ) (1/27) 13 ( ) (2/3) (2/17?) ( ) (2) / 44
4 ( ) (2) / 44
5 1 2 3 ( ) (2) / 44
6 (V, E) V E 2 V V 2 V = {1, 2, 3, 4, 5} E = {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 5}} {2, 5} = {5, 2} ( ) ( ) (2) / 44
7 V = {1, 2, 3, 4, 5} E = {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 5}} ( ) (2) / 44
8 G = (V, E) V G E G V G E G {u, v} E u, v v e v e u v u v V = {1, 2, 3, 4, 5} E = {{1, 2}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 5}} 2, 3 {2, 3} {2, 3} ( ) (2) / 44
9 ( ) (2) / 44
10 ( ) (2) / 44
11 G = (V, E) G I V 2 u, v I {u, v} E ( ) (2) / 44
12 ( ) 22 ( ) (2) / 44
13 22 ( ) (2) / 44
14 G = (V, E) v V σ v {0, 1} σ v = 0 v σv = 1 v σ v = 1 v V = ( ) (2) / 44
15 P 1 P 2 P 3 P 4 P 5 P n ( ) (2) / 44
16 n ( ) (2) / 44
17 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44
18 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44
19 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44
20 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44
21 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44
22 P 5 2 (A) = P 4 (B) = 2 P 3 + P 5 = P 4 + P 3 ( ) (2) / 44
23 ( ) P n 2 ( n 3) (A) = P n 1 (B) = 2 P n 2 + n 3 P n = P n 1 + P n 2 ( ) (2) / 44
24 ( ) P n 2 ( n 3) (A) = P n 1 (B) = 2 P n 2 + n 3 P n = P n 1 + P n 2 ( ) (2) / 44
25 ( ) P n 2 ( n 3) (A) = P n 1 (B) = 2 P n 2 + n 3 P n = P n 1 + P n 2 ( ) (2) / 44
26 a n = P n 2 (n = 1 ) a n = 3 (n = 2 ) a n 1 + a n 2 (n 3 ) ( ) (2) / 44
27 P n P 2 (G n ) G 1 G 2 G 3 G 4 G 5 G n ( ) (2) / 44
28 P n P 2 G n 2 (A) = (B) = G k ( ) (2) / 44
29 P n P 2 G n 2 (A) = (B) = G k ( ) (2) / 44
30 P n P 2 G n 2 (A) = (B) = G k ( ) (2) / 44
31 P n P 2 (H n ) H 1 H 2 H 3 H 4 H 5 H n n 2 G n = H n + H n 1 ( ) (2) / 44
32 P n P 2 H n 2 (A) = (B) = n 2 H n = G n 1 + H n 1 ( ) (2) / 44
33 P n P 2 H n 2 (A) = (B) = n 2 H n = G n 1 + H n 1 ( ) (2) / 44
34 P n P 2 H n 2 (A) = (B) = 2 + G n 1.. H n 1.. n 2 H n = G n 1 + H n 1 ( ) (2) / 44
35 P n P 2 H n 2 (A) = (B) = 2 + G n 1.. H n 1.. n 2 H n = G n 1 + H n 1 ( ) (2) / 44
36 P n P 2 b n = G n c n = H n b n = c n = { 3 (n = 1 ) c n + c n 1 (n 2 ) { 2 (n = 1 ) b n 1 + c n 1 (n 2 ) ( ) (2) / 44
37 1 2 3 ( ) (2) / 44
38 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end fnct(n) a ( ) (2) / 44
39 n a n a n a n a ( ) (2) / 44
40 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end f n = fnct(n) a ( ) (2) / 44
41 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end 2 n 1 a 4 5 ( ) (2) / 44
42 A 1: def fnct(n) 2: print "a" 3: if n > 2 4: fnct(n-1) 5: fnct(n-2) 6: end 7: end f n = { 1 (n 2 ) 1 + f n 1 + f n 2 (n 3 ) ( ) (2) / 44
43 Euclid Euclid 1: def gcd(a, b) # precondition: a >= b 2: print "G" 3: if b == 0 4: return a 5: else 6: gcd(b, a % b) 7: end 8: end a % b = a b gcd(a, b) G ( ) ( ) (2) / 44
44 Euclid (1) a b G ( ) (2) / 44
45 Euclid (2) a b G ( ) (2) / 44
46 Euclid Euclid 1: def gcd(a, b) # precondition: a >= b 2: print "G" 3: if b == 0 4: return a 5: else 6: gcd(b, a % b) 7: end 8: end g n = max {gcd(a, b) G } a 1,b n g n = b n g n ( ) (2) / 44
47 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44
48 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44
49 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44
50 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44
51 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44
52 Euclid a, b 1 a b a a % b 2 a = bq + r ( 0 r < b) a % b = r a b q = a r b r = 1 r b b b > 0 q q 1 a a b r < b 2 2 a a b > r = a bq a b < a 2 2 a r 2 a = 2 ( ) n n n 2 = n 2 ( ) (2) / 44
53 Euclid Euclid 1: def gcd(a, b) # precondition: a >= b 2: print "G" 3: if b == 0 4: return a 5: else 6: gcd(b, a % b) 7: end 8: end g n = gcd(a, b) G a, b... ( b = n) ( ) (2) / 44
54 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44
55 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44
56 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44
57 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44
58 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44
59 Euclid (1) g n = gcd(a, b) G = 1 + gcd(b, a % b) G a % b = 0 g n = 2 ( gcd(b, a % b) ) a % b 0 ( ) (2) / 44
60 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
61 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
62 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
63 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
64 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
65 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
66 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
67 Euclid (2) g n = gcd(a, b) G = 1 + gcd(b, a % b) G = gcd(a % b, b % (a % b)) G max a 1,b b/2 {gcd(a, b ) G } = 2 + g b/2 = 2 + g n/2 n 1 g n 2 + g n/2 { = 1 n = 0 g n 2 + g n/2 n 1 ( ) (2) / 44
68 Collatz 1: def collatz(n) 2: print n 3: if n % 2 == 0 4: collatz(n/2) 5: else 6: collatz(3*n+1) 7: end 8: end Collatz ( ) n collatz(n) 1 n (Oliveira e Silva 10) ( ) (2) / 44
69 1 2 3 ( ) (2) / 44
70 ( ) (2) / 44
71 ( ) (2) / 44
72 ( ) TA OK OK ( ) (2) / 44
73 1 2 3 ( ) (2) / 44
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