An Approach to Counting Odd Numbers

by Frank C. Fung - 1st published in October, 2008.

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Section VIII - A First Proposal on counting { Odd Numbers }

Summary for the Section :

• In this Section , we shall make a first proposal for counting { odd numbers } .

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The Proposal :

• Our proposal here is simply to do the counting in stages :
  • For { Stage 1 } , we count the { odd numbers } , by { categories } , up to [ 3 ] .

  • For { Stage 2 } , we count the { odd numbers } , by { categories } , up to [ 9 ] .

  • For { Stage 3 } , we count the { odd numbers } , by { categories } , up to [ 27 ] .

  • For { Stage 4 } , we count the { odd numbers } , by { categories } , up to [ 81 ] .

  • and so-on-and-so-forth .

  We then have this table below summarizing the [ count ] for { Stage 1 } thru { Stage 10 } :

  Value of N	Value of 3^N	* * *	Category											* * *	Total Count
  			Zero	I	II	III	IV	V	VI	VII	VIII	IX	X
  ******	******		******	******	******	******	******	******	******	******	******	******	******		******
  1	3		1	1	0	0	0	0	0	0	0	0	0		2
  2	9		1	3	1	0	0	0	0	0	0	0	0		5
  3	27		1	8	4	1	0	0	0	0	0	0	0		14
  4	81		1	21	14	4	1	0	0	0	0	0	0		41
  5	243		1	52	48	16	4	1	0	0	0	0	0		122
  6	729		1	128	148	66	17	4	1	0	0	0	0		365
  7	2,187		1	326	443	228	74	17	4	1	0	0	0		1,094
  8	6,561		1	846	1,303	758	274	77	17	4	1	0	0		3,281
  9	19,683		1	2,226	3,798	2,451	970	296	78	17	4	1	0		9,842
  10	59,049		1	5,967	10,948	7,805	3,316	1,083	305	78	17	4	1		29,525

• We simply note here that :
  • Once we have completed the [ counting ] for { Stage [ N ] } , then :
    • all { Category II } thru { Category [ N + 1 ] } numbers for the next stage , { Stage [ N + 1 ] } , may be determined ,

      so that new { Category I Odd-Numbers } for { Stage [ N + 1 ] } may also be determined .

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The Open-Ended Question :

• The Oped-Ended Question here is , of course :
  • Whether we can make some generalizations on our { Counting Procedure } ,
    • as we move forward to higher and higher { stages } .

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go to the next section : Section IX - Concluding Remarks

go to the last section : Section VII - Details on { Category VII / VIII / IX Odd-Numbers }

return to the { Counting Odd-Numbers } HomePage

original dated 2008-10-04