Friday, February 16, 2018

Octagonal and Triangular Numbers (1)

Some time ago when I was exploring the use of holistic (circular) notions for deriving perhaps novel new insights with respect to simple number relationships, I became fascinated with - what I referred to at the time as - self generating numbers.

So a self generating number refers to a number that repeats itself according to some well-defined procedure based on subtraction.

And I basically defined two types of such relationships linear and circular respectively.

So one linear type example (with hierarchy) results from the ordering of the digits in descending (largest digit 1st) and ascending (smallest digit 1st) respectively and then subtracting the latter from the former number. So when the original number results from this operation, we thereby have a self-generating number.

For example if we take the number 495 to illustrate when arranging its digits in descending order, we obtain 954 and then in ascending order 459.

And then when we obtain 954 – 459, the result is 495 (i.e. the original number).

Now in fact 496 represents the only example of a 3-digit number (in the denary system) that is self-generating in this sense, with 6174, the only example of a corresponding 4-digit number.

One other interesting feature of such numbers is that when divided by 9 that a palindrome (or close palindrome) frequently results.

So 495/9 = 55; 6174/9 = 686.

I also looked at the circular equivalent, which is especially well represented through cyclic primes.

Self generation is this sense results when the same circular sequence of digits is preserved through subtraction.

For example 142857 represents the well-known unique cyclic system of digits associated with the reciprocal of 7.

Then if we obtain the difference of any 6-digit numbers, that preserves the same cyclic order of these digits that the result will be a number (that likewise preserves the same cyclic order).

So for example, 428571 – 142857 = 285714 (which preserves the same cyclic order).

Note that if we adopt the initial linear order (already mentioned) that self-generation will not occur
So 875421 – 124578 = 750483 (which is neither self-generating in a linear or circular sense).

I then examined the simpler linear case (without hierarchy) where subtraction takes place with the order of digits simply reversed.

So for example taking 6174 again to illustrate its reverse is 4716

And the difference = 6174 – 4716 = 1458. So self-generation does not take place.

Now, perhaps surprisingly there does not appear to be any numbers (in base 10) that are self-generating (in this non-hierarchical sense).

So we have to look to other number bases for this feature to arise.

Fro example in base 8, if we take the 2-digit number, 275 and then subtract it from its reverse we get 572 – 275 = 275.  So this clearly then is a self-generating number in a linear (non-hierarchical) manner.

I then realised that a particularly interesting case would arise here in the 2-digit case as - by definition - both linear and circular self-generation would be involved. And also the hierarchical and non-hierarchical definitions would likewise coincide.

So the quest was on therefore to find all 2-digit self generating numbers that satisfied these requirements (in all relevant number bases).

And I found that a fascinating solution existed. So in general terms if n represents the number base, the relevant bases where self-generation occurs is given by 2 + 3k (where k = 0, 1, 2, 3, …)

And if a represents the 1st digit of the corresponding self-generating number, its value is given as k with the second digit = 2k + 1.

Therefore when k = 0 the relevant number base = 2. The 1st digit = 0 and the second digit = 1

Therefore in base 2, 01 is a self generating number (in both a linear and cyclical sense).

So 10 – 01 = 01.

Then when k = 1, the next relevant number base = 5. Therefore the 1st digit = 1, and the 2nd = 3.

So in base 5, 13 is a self-generating number in the manner defined.

So 31 – 13 = 13.

When k = 2, the next number base = 8. The 1st digit = 2, and the 2nd digit = 5.

So 52 – 25 = 25.

And giving just one more example to illustrate, the next number base is 11 with the 1st digit = 3 and the 2nd digit = 7.

So 73 – 37 = 37 (in base 11).   

Now if one converts all these self-generating numbers into conventional denary (base 10) format, the resulting sequence arise

i.e. 1, 8, 21, 40, …, which represents the well-known octagonal numbers.

Thus the octagonal numbers in this sense are uniquely related to all these 2-digit self-generating numbers (in other number bases).

One other fascinating feature of these numbers struck me at the time i.e. that they represent the cyclic digit sequence (in their respective number bases) of the reciprocal of 3.

So for example if we were to express 1/3 in base 8, we would obtain .252525…

Thus the uniquely recurring digits here are 25.

In one way, this is a remarkable feature, which connects the octagonal and triangular numbers 1, 3, 6, 10, … in a way that is perhaps not commonly realised.

And when we look more closely at the triangular, striking connections can indeed be found with the octagonal numbers.

Now the triangular numbers repeat naturally groups of 3 where the 1st member of the group leaves a remainder of 1 (when divided by 3) while 3 is always a factor of the other two members of the group.

So taking the 1st 3 members 1, 3 and 6 if we now add the 3 members of the group and subtract 1, the total will then be divisible by 3. So {(1 + 3 + 6) – 1}/3 = 9/3 = 3 (which is the 2nd triangular number).

Then we treat the next group of 3 in the same manner, we obtain {(10 + 15 + 21) – 1}/3 = 45/3 = 15 (which is the 5th triangular number).

Then with the next group of 3 we have {(28 + 36 + 45) – 1}/3 = 108/3 = 36 (which is the 8th triangular number).

So the ordinal ranking of the triangular numbers obtained in this manner exactly match the corresponding number bases (where the self generating numbers arise).

And again the octagonal numbers are then the representation of these numbers in base 10.

And a more direct connection can be made.

If we start with the 3rd triangular number and then divide by 3 we obtain 6/3 = 2.

If we now keep increasing the ordinal rank by 6 and then dividing (the number arising) by the corresponding cardinal number we will generate the sequence 2, 5, 8, 11, … which matches in cardinal fashion the ordinal rankings of the number bases in which the self generating numbers arise.

So the 9th triangular number = 45 and 45/9 = 5; the 15th cardinal number = 120 and 120/15 = 8; the 21st triangular number = 231 and 231/21 = 11 and so on.

Therefore in a way it is very striking how such connections can be shown as between number (with respect to both cardinal and ordinal usage) both within the denary base and across other relevant number bases.

And this in turn relates to the very nature of the self-generating numbers (on which these connections are based) where both linear and circular notions coincide.

Again the octagonal numbers are:

1, 8, 21, 40, 65, 96, 133, ....

And the triangular numbers are:

1, 3,  6, 10, 15, 21,  28, ...

And if we subtract each successive term in the triangular from the corresponding term in the octagonal, we obtain,

0, 5, 15, 30, 50, 75, 105

= 5 (1, 3, 6, 10, 15, 21, ...)

And the sequence within the brackets is once again that of the triangular numbers.

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