sw00017math-nnss

The nnss: Natural Number Summation Sequence


Yesterday I found a site where modulo is
explained clearly enough so that I was finally
able to understand the concept of it and how it is
used.  Also yesterday some new insights regarding
the natural number summation sequence appeared in
my consciousness.  I am going to try to clearly
present them along with my older insights, and I do
not expect I will attempt to use modular arithmetic.
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If one allows "0" (accepting it as a non-negative
integer) to be the first term of the natural number
sequence/ so that it is at term-position 1 in that
sequence, one will get these results:


0      1 x 0 = 0 and 0 + 1 = 1
1      2 x .5 = 1 and 1 + 2 = 3
2      3 x 1 = 3 and 3 + 3 = 6
3      4 x 1.5 = 6 and 6 + 4 = 10
4      5 x 2 = 10 and 10 + 5 = 15
5      6 x 2.5 = 15 and 15 + 6 = 21
6      7 x 3 = 21 and 21 + 7 = 28
7      8 x 3.5 = 28 and 28 + 8 = 36
8      9 x 4 = 36 and 36 + 9 = 45

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The first number in each row is the natural 
   number.
The second number in each row is the term-position
   number.
The third number in each row is the t-p # divided
   by 2, and then made less by .5.
The fourth number in each row is the value of the
   nnss term at that term's position.
The fifth number in each row is the value of the
   nnss term at that term's position.
The sixth number in each row is the term-position
   number.
The seventh number in each row is the value of the
   next greater nnss term.
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My approach to modular arithmetic has been to work
around it.  It is known that multiples of "8" are
central to determining the odd squares.  What I saw
was that/ adding 1 to (8 times any number in the 
nnss) = an odd square so long as "0" is allowed to
be the first term in the nnss.
   Example: 8 x 36 = 288   288 + 1 = 289   17 x 17
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Letting non-negative integer = natural number,
I also saw that by dividing a term position
in a (0, 1, 2, 3, 4, 5, 6, 7, 8, n) natural number
sequence by 2, and then multiplying that quotient
by the value of the non-negative integer at that
position would net one the sum of the non-negative 
integers through the natural numberat that  
position.
          Example: 3/2 = 1.5   1.5 x 2 = 3
(note: "3" is 2's term position.)
#
Further examples:  If you divide any non-negative
integer by 2, then minus .5 from that quotient,
then multiply that remainder by the non-negative
integer you have chosen, the resulting product
will be the natural number summation sequence term
for all the non-negative integers prior to the
non-negative integer you have chosen.
  If after having divided your chosen non-negative
integer by 2, you add .5 to that quotient and then
multiply that sum by your chosen non-negative 
integer, the resulting product will be the natural
number summation sequence term for all the non-
negative integers including/ your chosen non-
negative integer.  Okay:
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100/2 = 50   50 - .5 = 49.5   100 x 49.5 = 4950
100/2 = 50   50 + .5 = 50.5   100 x 50.5 = 5050
101/2 = 50.5   50.5 - .5 = 50   101 x 50 = 5050
101/2 = 50.5   50.5 + .5 = 51   101 x 51 = 5151
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(The ghost of Gauss is in all of this.)
# 

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Another point of interest is:

 1 = 1 x 1
 3 = 1 x 3
 6 = 2 x 3
10 = 2 x 5
15 = 3 x 5
21 = 3 x 7
28 = 4 x 7 = 2 x 14
36 = 4 x 9 = 3 x 12 = 2 x 18
45 = 5 x 9 = 3 x 15
55 = 5 x 11


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 2006-11-11
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 [ last modified: 2007-10-17 ]           


Brian A. J. Salchert