Octagonal and Triangular Numbers (2)

The octagonal numbers i.e.

1, 8, 21, 40, 65, 96, …have built in to their numerical structure some very interesting relationships.

1 =   1 * 1,

8 =   2 * 4,

21 = 3 * 7,

40 = 4 * 10

65 = 5 * 13

96 = 6 * 16

      …

So the kth term i.e. t_k = k(3k – 2) = 3k² – 2k

Also when each term is multiplied by 3, it results in a number 1 less than a perfect square.

So 1 * 3 =  3  =  2² – 1

     8 * 3  = 24 = 5²  – 1

   21 * 3  = 63 = 8² – 1

             …

So once again we see the relationship between 3 and the the sequence 2, 5, 8, … (in which bases we have the unique digit sequence of the reciprocal of 3).

So 3 * t_k  = (3k – 1)² – 1; so t_k = {(3k – 1)² – 1}/3 = 3k² – 2k

Also each term can be expressed as the difference of two squares.

So 1 =  1² – 0²

       8  = 3² – 1²

   21  = 5² – 2²

    40  = 7² – 3²

                 …

So again t_k = (2k – 1)² – (k – 1)²  = k(3k – 2) = 3k² – 2k.

We have already shown the connection between the octagonal self-generating numbers (with both non-hierarchical linear and circular features).

Now the coincidence of both linear and circular self-generating features can only apply when numbers have two digits.

However the non-hierarchical linear aspect can be continued into numbers with more than 2 digits.

Quite simply when a 2-digit number is self-generating in this non-hierarchical sense then a n-digit number (where n > 2) is likewise self-generating where the additional digits are 1 less than the base in question and inserted between the 1st and 2^nd digits (of the 2-digit number).

So for the simplest case, in base 2, 01 is self-generating (as 10 – 01 = 01).

This therefore entails that 3-digit number 011 is also self-generating (where the additional digit inserted is 1 less than the corresponding number base 2).

So 110 – 011 = 011  i.e. 3 in denary terms. 

So in the next relevant number base 5, 143 is self generating, where 341 – 143 = 143, i.e. 48 in denary terms.

In the next relevant number base 8, 275 is self-generating, where 572 – 275 = 275, i.e. 189 in denary terms.

Then in the next relevant number base 11, 3A7 is self-generating as 7A3 – 3A7 = 3A7, i.e. 480 in denary terms.

Now if we look at the denary nature of the sequence generated 3, 48, 189, 480, …, we can again show its links to the octagonal numbers

So, 3 = (1 * 3) *  1

    48 = (2 * 3) *  8

  189 = (3 * 3) * 21

  480 = (4 * 3) * 40  

              …

And the sum of the 1^st n terms of the sequence 1, 2, 3, 4, … is the nth triangular number.

So the sum of the first 4 terms i.e. 1 + 2 + 3 + 4 = 10 i.e. the 4^th triangular no.

Then when we move into 4-digit numbers 0111 is self-generating with respect to the base 2 i.e. 7 in denary terms.

Next 1443 is self-generating with respect to base 5 i.e. 248 in denary terms.

Then 2775 is self-generating with respect to base 8 i.e. 1533 in denary terms.

Finally to illustrate 3AA7 is self-generating with respect to base 11 i.e. 5320 in denary terms.

Again a fascinating connection can be shown with the octagonal numbers.

So   7     =     7  * 1

248   =   31 * 8

1533 =   73 * 21

5320 = 133 * 40

                 …

Now the other sequence of numbers 7, 31, 73, 133, … is fascinating in its own right

7 = 6 + 1, 31 = 30 + 1, 73 = 72 + 1 133 = 132 + 1, …

And 6 =  6 *  1

      30 =  6 *  5

      72 =  6 * 12

     132 = 6 * 22

           …

And 1, 5, 12, 22, … are the pentagonal numbers, which in turn are related to the triangular numbers.

The average of the 1^st n pentagonal numbers is the nth triangular no.

So the average for example of the first four pentagonal numbers = 40/4 = 10 (i.e. the 4^th triangular number).