2005 0 (a + b)(a b) = (a + b)a (a + b)b = aa + ba ab bb = a 2 b 2 (a + b)(a b) a 2 b 2 2 ( )( + ) = ( + ) ( + ) = ( + ) ( + 2 ) = + ( ) 2 = 2 (a + b)(a b) = a 2 b 2 ( )( + ) ( + ) ( + + 2 ) http://www.hyuki.com/story/genfunc.html http://www.hyuki.com/ Hiroshi Yuki c 2005, All rights reserved.
( )( + + 2 ) = ( + + 2 ) ( + + 2 ) = ( + + 2 ) ( + 2 + 3 ) = + ( ) + ( 2 2 ) 3 = 3 ( )( + ) = 2 ( )( + + 2 ) = 3 ( )( + + 2 + 3 ) = 4 ( )( + + 2 + 3 + 4 ) = 5 ( )( + + 2 + 3 + 4 + + n ) = n+ ( ) ( ) n k = n+ n n 0 n = 0 0 ( ) k = n ( ) k = n+ 0 0 n k = n+ 2
n 0 0 n r, r, r 2, r 3,..., r n,... 0 n + r + r 2 + r 3 + + r n = n r k = rn+ r n r n r < n rk + r + r 2 + r 3 + = lim n r k n! r n+ = lim n! r = r r < r n+ 0 n lim n! a k a k r k = r r < r r 3
+ + 2 + 3 +,,,,...,,,,... + + 2 + 3 +,,,,... + + 2 + 3 + + + 2 + 3 + =,,,,..., r, r 2, r 3,... r k k, r, r 2, r 3,... + r + r 2 2 + r 3 3 + + r + r 2 2 + r 3 3 + + + 2 + 3 + = r + r + r 2 2 + r 3 3 + = r, r, r 2, r 3,... r a 0, a, a 2, a 3,... a 0 + a + a 2 2 + a 3 3 + 4
2 a k 0,,, 2, 3, 5, 8,... 2 0,, 0 + =, + = 2, + 2 = 3, 2 + 3 = 5, 3 + 5 = 8,... a k a 0 0 a k 2 a k = a k 2 + a k 0 (k = 0) a k = (k = ) a k 2 + a k (k 2) 2 a k k a k k 000 0,,, 2, 3, 5, 8,... a k k f() a 0, a, a 2, a 3,... f() 5
f() f() = a 0 0 + a + a 2 2 + a 3 3 + a 4 4 + a 5 5 + a 6 6 + = 0 0 + + 2 + 2 3 + 3 4 + 5 5 + 8 6 + = + 2 + 2 3 + 3 4 + 5 5 + 8 6 + f() a k f() a k 2 + a k = a k 3 f() = + a k 2 k 2 + a k k + a k k + a k 2 a k 3 a k 2 k 2 a k 2 k 2 2 a k 2 k 2 = a k 2 k a k k = a k k 0 a k 0 k 0 = a k 0 k f() 2,, 0 : 2 f() = a 0 2 + a 3 + a 2 4 + a 3 5 + a 4 6 + a 5 7 + a 6 8 + 2: f() = a 0 + a 2 + a 2 3 + a 3 4 + a 4 5 + a 5 6 + a 6 7 + 3: 0 f() = a 0 0 + a + a 2 2 + a 3 3 + a 4 4 + a 5 5 + a 6 6 + + 2 3 6
( 2 + 0 )f() = a 0 0 + a 0 a }{{ +(a } 0 + a a 2 ) 2 + + (a k 2 + a k a k ) k + 0 a k 2 + a k a k 0 0 a 0 = 0 a = ( 2 + )f() = f() f() f() = 2 0,,, 2, 3, 5, 8,... 2 f() f() 2 2 k k a k 2 = + + 2 + 3 + r = + r + r2 2 + r 3 3 + 2 r 2 7
4 2 r 2 r 2 2 2 = ( r)( s) r, s r + s 2 r, s r + s = s ( r)( s) + r ( r)( s) 2 (r + s) = (r + s) + rs 2 = r, s 2 2 R, S, r, s R r + S s R r + S s = R( s) ( r)( s) + S( r) ( r)( s) (R + S) (rs + sr) = (r + s) + rs 2 2 R + S 8
(R + S) (rs + sr) (r + s) + rs 2 = 2 R + S = 0 rs + sr = r + s = rs = 4 4 R S r s R = r s, S = r s f() r, s f() = ( r)( s) = R r + S s = ( r s r ) s = ( ) ( + r + r 2 2 + r 3 3 + ) ( + s + s 2 2 + s 3 3 + ) r s = r s ( ) (r s) + (r 2 s 2 ) 2 + (r 3 s 3 ) 3 + f() = r k s k r s k r, s a k = rk s k r s r, s r s { r + s = rs = r, s 2 (r + s) + rs = 0 r, s 9
2 = 0 r > s r s = 5 = ± 5 2 r = + 5 2 s = 5 2 r k s k r s = ( + 5 2 ) k ( 5 5 2 ) k = ( + 5) k ( 5) k 2 k 5 a k a k = ( + 5) k ( 5) k 2 k 5 5 k = 0,, 2, 3, 4 a 0 = ( + 5) 0 ( 5) 0 2 0 5 a = ( + 5) ( 5) 2 5 a 2 = ( + 5) 2 ( 5) 2 2 2 5 a 3 = ( + 5) 3 ( 5) 3 2 3 5 a 4 = ( + 5) 4 ( 5) 4 2 4 5 = 0 = 2 5 2 5 = = 4 5 4 5 = = 6 5 8 5 = 2 = 48 5 6 5 = 3 0
0,,, 2, 3 k 5 k a k f() f() f() k 0
Graham, Knuth, Patashnik,, ISBN 4-320-02668-3,, 993.,, ISBN 4-535-7828-4,, 2002. 6 (generating function) 0 PDF URL http://www.hyuki.com/story/genfunc.html Copyright (C) 2005 Hiroshi Yuki ( ) All rights reserved. 2005 0 7 2005 0 8 2