We have
seen that the nth term of the octagonal series, i.e.

1, 8, 21,
40, 65, …,

is given by
3n

^{2 }– 2n where n = 1, 2, 3, …However what is interesting is that a “shadow” series (comprising positive valued terms) can be obtained by letting n = 0, – 1, – 2, …, i.e.

0, 5, 16, 33, 56, …

And again the same neat numerical
features that characterise the octagonal series, also characterise its “shadow”
referred to in OEIS as A045944 (The Rhombic Matchstick Numbers).

So when we multiply each of the
numbers of this sequence by 3, we obtain a number that is 1 less than a perfect
square.

So 0 * 3 = 0 = 1

^{2}– 1; 5 * 3 = 15 = 4^{2 }– 1; 16 * 3 = 48 = 7^{2 }– 1 and so on.
Likewise each of these numbers can
be expressed as the difference of two squares.

So 0 = 1

^{2}– 1^{2}; 5 = 3^{2}– 2^{2}; 16 = 5^{2}– 3^{2}and so on.
And Also in the case of the
octagonal series

0 = 0 * 2 1 =
1 * 1

5 = 1 * 5 8 =
2 * 4

16 = 2 * 8 21 = 3 * 7

33 = 3 * 11 40 = 4 * 10

… …

So we can see that the number
bases that were so relevant in the case of self generating numbers and the
octagonal series i.e. 2, 5, 8, 11, …, now naturally appear with this “shadow”
sequence.

Likewise in complementary fashion
the number bases applicable to self generating numbers in the “shadow” sequence
i.e. 1, 4, 7, 10, …naturally appear in the octagonal series.

In fact because the value of the
respective numbers bases is directly related to the value of k it is the
negative number bases that are relevant with respect to the “shadow” sequence.

In other words the self generating
numbers that arise in the negative number bases i.e.

– 1, – 4, – 7, – 10,…, in fact are
the respective terms 0, 5, 16, 33, … of the shadow sequence.

In fact the 1st case is trivial
for 00 in base – 1 is 00 and 00 – 00 =
00.

However it is somewhat trickier in
subsequent number bases.

In fact in base – 4, 5 would be
written as – 23.

Now if we look at the right hand
digit this indeed represent 3 units. However the 2 in base – 4 = 2 * (– 4) = –
8.

Therefore 23 in base – 4 = – 8 + 3 = – 5 (in
denary terms).

Therefore 5 (in base 10) = – 23
(in base – 4).

And the reverse of – 23 = – 32 (in
base – 4).

The difference of the two = – 32 –
(– 23) = – 32 + 23

And 32 = 3 * (– 4) + 2 = – 12 + 2 = – 10 (in
denary terms)

Therefore – 32 = 10 and 23 = – 5.

So – 32 + 23 = 10 – 5 = 5.

Therefore when we subtract –
23 from its reverse in base – 4, we get – 23.

So – 23 is self-generating (in
both cyclical and linear non-hierarchical terms) in this base.

Then next case arises in base – 7,
where – 35 (= 16 in base 10) is now the relevant number.

And – 53 – (– 35 ) in base –
7, =
– 53 + 35 = 32 – 16 = 16 (in base 10).

So – 35 is self-generating in base
– 7.

And just as the two digits of the
self-generating numbers in the positive bases 2, 5, 8, 11, … represent the
unique digit sequence of the reciprocal of the cyclic prime 3 in these bases,
likewise the two digits in the negative bases – 1, – 4, – 7, – 10 again
represent the unique digit sequence of the reciprocal of 3.

Now if we line up the octagonal
and its corresponding “shadow” sequence as follows, we have,

1, 8, 21, 40, 65, …

0, 5, 16, 33, 56, …

Then subtracting each term of the
latter “shadow” from the corresponding term of the former octagonal sequence we
get,

1, 3, 5, 7, 9, …

In other words, we have the
sequence of odd-numbered natural integers.

This for example using Leibniz’s
well- known formula, we can express π in terms of the octagonal and shadow
sequences.

So π/4 = 1/(1 – 0) – 1/(8 – 5) +
1/(21 – 16) – 1/(40 – 33) + …

It is also fascinating how we can
derive the natural number sequence, through dividing each successive term of
one sequence by the corresponding successive number base of the complementary
sequence.

Thus with respect to the “shadow”
sequence the terms are 0, 5, 16, 33, ..

and the corresponding number bases
(related to self-generating numbers) of the octagonal sequence are 2, 5, 8, 11,
…

And 0/2 = 0; 5/5 = 1; 16/8 = 2;
33/11 = 3 and so on.

Then with respect to the octagonal
sequence, the terms are 1, 8, 21, 40, … and the corresponding number bases
(related to self-generating numbers) of the “shadow” sequence are – 1, – 4, –
7, – 10, …

And 1/– 1 = – 1; 8/– 4 = – 2; 21/–
7 = – 3; 40/– 10 = – 4 and so on.

If we now combine the positive
number bases i.e. 2, 5, 8, 11, … directly with the corresponding terms of the
octagonal series, a fascinating pattern emerges.

So
2 * 1 = 2

5 *
8 = 40

8 * 21
= 168

11 * 40 = 440

…

Now the
sum of the first (i.e. one) term = 2 = 1 * 2

The sum
of the first two terms = 2 + 40 =
42 = 6 * 7

The sum
of the first 3 terms = 42 + 168 = 210
= 14 * 15

The sum
of the first 4 terms = 210 = 440 = 650
= 25 * 26

So the
cumulative total of terms always can be expressed as the product of two
consecutive numbers.

This also applies to the shadow
sequence, multiplied by the number bases,
1, 4, 7, 10, …

So
1 * 0 = 0

4 *
5 = 20

7
* 16 = 112

10 * 33 =
330.

So the cumulative total of 1 = 0 =
0 * 1

The cumulative total of 2 = 20 = 4 * 5

The cumulative total of 3 = 132 =
11 * 12

The cumulative total of 4 = 462 =
21 * 22

Notice how the increase of each
number in the consecutive sequence is related to the number bases. So each
number increases by 4, then 7, then 10 and so on whereas with octagonal numbers
each number (in the consecutive sequence).

Finally, we can again make a connection
with the triangular nos. in the following way.

Again, if we line up the octagonal
and its corresponding “shadow” sequence as follows, we have,

1, 8, 21, 40, 65, …

0, 5, 16, 33, 56, …

Then adding each respective term
we get,

1, 13, 37, 73, 121, …

Then subtracting 1 we get

0, 12, 36, 72, 120, …

= 12 (0, 1, 3, 6, 10, …)

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