Overview  Package   Class  Tree  Deprecated  Index  Help 
 PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    
SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD

tribble.math.misc
Class MapNtoQ

java.lang.Object
  extended by tribble.math.misc.MapNtoQ

public class MapNtoQ
extends java.lang.Object

Generates a sequence of rational numbers. Sequence S(n) is defined as: 

    S(0) =    0
    S(2n) =   S(n)+1, for n > 0
    S(2n+1) = 1/S(n)+1 = 1/S(2n)

Sequence S(n) maps every natural number to every rational number. In other words, S(n) is a bijection between all n > 0 in N to all r = p/q in Q.

A partial list of S(n):
    0. 0/1 = 0    20. 7/3        40. 10/3        60. 13/5        80. 13/3
    1. 1/1 = 1    21. 3/7        41. 3/10        61. 5/13        81. 3/13
    2. 2/1 = 2    22. 7/4        42. 10/7        62. 13/8        82. 13/10
    3. 1/2        23. 4/7        43. 7/10        63. 8/13        83. 10/13
    4. 3/1 = 3    24. 7/2        44. 11/4        64. 7/1  = 7    84. 17/7
    5. 1/3        25. 2/7        45. 4/11        65. 1/7         85. 7/17
    6. 3/2        26. 7/5        46. 11/7        66. 7/6         86. 17/10
    7. 2/3        27. 5/7        47. 7/11        67. 6/7         87. 10/17
    8. 4/1 = 4    28. 8/3        48. 9/2         68. 11/5        88. 15/4
    9. 1/4        29. 3/8        49. 2/9         69. 5/11        89. 4/15
   10. 4/3        30. 8/5        50. 9/7         70. 11/6        90. 15/11
   11. 3/4        31. 5/8        51. 7/9         71. 6/11        91. 11/15
   12. 5/2        32. 6/1 = 6    52. 12/5        72. 13/4        92. 18/7
   13. 2/5        33. 1/6        53. 5/12        73. 4/13        93. 7/18
   14. 5/3        34. 6/5        54. 12/7        74. 13/9        94. 18/11
   15. 3/5        35. 5/6        55. 7/12        75. 9/13        95. 11/18
   16. 5/1 = 5    36. 9/4        56. 11/3        76. 14/5        96. 11/2
   17. 1/5        37. 4/9        57. 3/11        77. 5/14        97. 2/11
   18. 5/4        38. 9/5        58. 11/8        78. 14/9        98. 11/9
   19. 4/5        39. 5/9        59. 8/11        79. 9/14        99. 9/11

Some relations derived for S(n):
   r1. S(n) for even n > 1 = S(n/2)+1.
   r2. S(n) for odd  n > 1 = 1/S(n-1).
   r3. S(2k) = k+1, for k >= 0.
   r4. S(n) = p/q, p > q for all even n > 0.
   r5. S(n) = p/q, p < q for all odd n.

Some inverse relations for S(n):
   i1. S-1(0) = 0.
   i2. S-1(1) = S-1(p/p) = 1, for p > 0.
   i3. S-1(p) = S-1(p/1) = 2p-1, for p > 0.
   i4. S-1(p/q) = 2S-1(p/q -1), if p > q.
   i5. S-1(p/q) = S-1(q/p) + 1, if p < q.

Since:
2006-04-05
Version:
$Revision: 1.1 $ $Date: 2006/04/06 00:42:16 $
Author:
David R. Tribble (david@tribble.com)

Constructor Summary
MapNtoQ()
           
 
Method Summary
static void main(java.lang.String[] args)
          Generate a sequence of S(n) numbers.
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

MapNtoQ

public MapNtoQ()
Method Detail

main

public static void main(java.lang.String[] args)
                 throws java.lang.Exception
Generate a sequence of S(n) numbers.

Usage

java MapNtoQ [[first_n] last_n]

Throws:
java.lang.Exception
Since:
1.1, 2006-04-05
Overview  Package   Class  Tree  Deprecated  Index  Help 
 PREV CLASS   NEXT CLASS FRAMES    NO FRAMES    
SUMMARY: NESTED | FIELD | CONSTR | METHOD DETAIL: FIELD | CONSTR | METHOD