This is another revisit to the natural number summation sequence and its relation to "8". Excluding "0", the first twelve nnss terms are: 1 3, 6 10, 15 21, 28 36, 45 55, 66 78 These can be approached variously; but how even squares are progenitors of them, and how odd squares are derived from them is valuable knowledge. As can be seen, this sequence can be shown in 2-term groups because the first two are odd integers and the second two are even integers, and so on in that alternating manner. Things to note: 1 + 3 = 4; 6 + 10 = 16; 15 + 21 = 36; 45 + 55 = 100; and so on in that manner. Obviously, the sums are even squares, meaning that hidden in any even square are two terms of the natural number summation sequence. Therefore, if 12 x 12 (or 144), is chosen for inspection, how are the nnss terms in it exposed? If 144 is divided by 2, the quotient is 72. If 12 is divided by 2, the quotient is 6. 72 - 6 = 66, and 72 + 6 = 78. As per above, 66 and 78 constitute a group of two even terms in the natural number summation sequence (nnss). - How do odd squares relate to this? Every odd square is 1 > any nnss term (including "0") multiplied by "8". So, say the target is 13 x 13 (or 169). How is it discovered? One way is to subtract "1" from "169" and then divide "168" by "8". Having done this, the next question is: What is the proof that "21" is an nnss term? There are several ways, but I want to try something. In the nnss there are odd groups and even groups. Since the first group is an odd group, I'm calling the odd groups the minor sequence, making the even groups the major sequence. Now 1 + 3 = 4, so "4" is a number which can be used as a divisor. Here's a run: .25, .75; 3.75, 5.25; 11.25, 13.75; 22.75, 26.25 Here's a run using even nnss terms: 1.5, 2.5; 7, 9; 16.5, 19.5; 28, 36 I'm sure you see the difference and also the nuances. 4 into 21 = 5.25, but--. .25 represents "1"; therefore, if 5.25 is divided by .25, the quotient should be 21, and it is. "21" is the sixth term in the nnss, which means that it is the sum of 1 + 2 + 3 + 4 + 5 + 6. If term position (tpo) is used as a guide, the results of dividing each term by its position creates an incontrovertible sequence: 1, 1.5, 2, 2.5, 3, 3.5, n Each of these numbers can also be obtained by adding "1" to an nnss tpo and then dividing by 2. From this it is also possible to get back to an nnss term's value by subtracting .5 from the (tpo + 1)/2 quotient and then multiplying that by the (tpo + 1). 3.5 - .5 = 3, and 3 x 7 = 21. [ By these methods any unknown nnss term can be found. Example: Let 63 be the tpo for an nnss term. 63 + 1 = 64. 64/2 = 32. 64 x 31.5 = 2016. ] So sayeth my calculator. What then is this?: (8 x 2016) + 1. 16129. What is the sqrt of that? Three calculator tries got me "127". - 3² 5² 7² 9² 11² 13² 15² 17² 19² 21² 23² 25² Now, the square root (sqrt) of 4 is 2; of 16 is 4; of 36 is 6; of 64 is 8; of 100 is 10; of 144 is 12 and 2 x 2 = 4, 2 x 4 = 8, 2 x 6 = 12, 2 x 8 = 16, 2 x 10 = 20; 2 x 12 = 24 Remember 1 + 3 = 4, 6 + 10 = 16, 15 + 21 = 36, 28 + 36 = 64, 45 + 55 = 100; 66 + 78 = 144 and 4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, 4 x 4 = 16, 4 x 5 = 20; 4 x 6 = 24 So, if the 2-term group position is 32, the integer between the odd square roots of each term in that group must be 4 x 32, which is 128. See "127" above. If the spread between the nnss terms in a group is known, then that value divided by 2 will equal that group's position (gpo). Therefore, (32 x 2) + 2016 must equal a term whose value is 2080; and (8 x 2080) + 1 must equal an odd integer which is 129 x 129. According to my calculator/ both the former and the latter is 16641. Of course, then, the difference between the odd squares divided by the spread between the related nnss terms will always equal "8". #

Brian A. J. Salchert

is a tiny wandering imaginary dinosaur which migrated from AOL in October of 2008.

## About Me

- brian (baj) salchert
- Rhodingeedaddee is my node blog. See
*my other blogs*and*recent posts*.

## Guide

[6-16-2009 Update Insert: Most of what is in this space is now moot. I found out what I was doing wrong and have reinstated Archives and Labels searches. They do work. However, in certain cases you may prefer Labels to Archives. Example: 1976 Today begins in November of 2006 and concludes in December of 2006, but there are other related posts in other months. Note: Labels only shows 20 posts at a time. There are 21 hubs, making 21 (which is for 1976 Today) an older hub.] ********************************* to my online poems and song lyrics using Archives. Use hubs for finding archival locations but do not link through them. Originally an AOL Journal, where the archive system was nothing like the system here, this blog was migrated from there to here in October of 2008. Today (Memorial/Veteran's Day, May 25, 2009) I discovered a glitch when trying to use a Blogger archive. Now, it may be template-related, but I am unable to return to S M or to the dashboard once I am in the Archives. Therefore, I've decided on this approach: a month-by-month post guide. The sw you see in the codes here stood for Salchert's Weblog when I began it in November of 2006. It later became Sprintedon Hollow. AOL provided what were called entry numbers, but they weren't consistent, and they didn't begin at the first cardinal number. That is why the numbers after "sw" came to be part of a post's code. ************** Here then is the month-by-month post guide: *2006* November: 00001 through 00046 - December: 00047 through 00056 -- *2007* January: 00057 through 00137 - February: 00138 through 00241 - March: 00242 through 00295 - April: 00296 through 00356 - May: 00357 through 00437 - June: 00438 through 00527 - July: 00528 though 00550 - August: 00551 through 00610 - September: 00611 through 00625 - October: 00626 through 00657 - November: 00658 through 00729 - December: 00730 through 00762 -- *2008* January: 00763 through 00791 - February: 00792 through 00826 - March: 00827 through 00849 - April: 00850 through 00872 - May: 00873 through 00907 - June: 00908 through 00931 - July: 00932 through 00955 - August: 00956 through 00993 - September 00994 through 01005 - October: 01006 through 01007 - November: 01008 through 01011 - December: 01012 through 01014 -- *2009* January: 01015 through 01021 - February: 01022 through 01028 - March: 01029 through 01033 - April: 01034 through 01036 - May: 01037 through 01044 - ******************************************************* 1976 Today: 2006/11 and 2006/12 -- Rooted Sky 2007: 2007/01/00063rsc -- Postures 2007: 2007/01/sw00137pc -- Sets: 2007/02/sw00215sgc -- Venturings: 2007/03/00216vc -- The Undulant Trees: 2007/03/00266utc -- This Day's Poem: 2007/03/00267tdpc -- Autobio: 2007/04/sw00316ac -- Fond du Lac: 2007/04/00339fdl -- Justan Tamarind: 2007/05/sw00366jtc -- Prayers in December: 2007/05/sw00393pindc -- June 2007: 2007/06/sw00440junec -- Seminary: 2007/07/sw00533semc -- Scatterings: 2008/08/00958sc ** Song Lyrics: 2008/02/sw00797slc ********** 2009-06-02: Have set S M to show 200 posts per page. Unfortunately, you will need to scroll to nearly the bottom of a page to get to the next older/newer page.

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## Saturday, May 3, 2008

### sw00875math--even-squares-to-odd-squares

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