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I offer here an initial development of some ideas and sequences I hope
you'll like. I'm interested in your comments, and especially in
seeing where you might take all this...<br>
<br>
<br>
<font size=3>The exposition here attempts to convey, using concrete
examples, a general idea I like to refer to as "numbrals”, and
suggests an approach for generating and describing many novel
sequences. The strategy is simply to systematically survey various
numbral systems we construct, using analogies drawn from regular number
theory concepts (eg addition, multiplication, partitions, divisors
etc). [Except where noted otherwise, all “A-numbers” here are
pending acceptance by the OEIS].<br>
<br>
<br>
Let’s start with some numbral partitions. Say we are working with
pairs of non-negative integers (i,j), and we define addition of pairs via
component addition:<br>
<br>
(i,j) + (k,l)
= (i+k,j+l).<br>
<br>
We can count the number of ways we can partition such a pair into
non-zero pairs, much like we count regular partitions. For example
there are four ways to partition the pair (2,1):<br>
<br>
(2,1)
(2,0)+(0,1) (1,0)+(1,0)+(0,1) (1,0)+(1,1)<br>
<br>
We can arrange these pair partition counts in an (infinite) numeric
table, T1:<br>
<br>
T1 | 0
1 2
3 4
5 6
7 8 9 ...<br>
---+-------------------------------------------------------------<br>
0 | 1
1 2
3 5 7
11 15 22 30<br>
1 | 1
2 4 7
12 19 30
45 67 97<br>
2 | 2
4 8 15
26 43 69 107
163 242<br>
3 | 3
7 15 28
49 81 130 203
311 466<br>
<font size=3>4 | 5
12 26 49
86 143 231 362
558 841<br>
5 | 7
19 43 81 143
236 381 595 916 1380<br>
6 | 11 30
69 130 231 381
616 961 1480 2230<br>
7 | 15 45
107 203 362 595 961
1492 2290 3440<br>
8 | 22 67
163 311 558 916 1480
2290 3506 5251<br>
9 | 30 97
242 466 841 1380 2230
3440 5251 7838<br>
<br>
where the number of partitions of (i,j) is given by the value entered at
the intersection of the row labeled i and column labeled j (starting with
i,j=0).<br>
<br>
Such a table is conventionally converted into a sequence by reading off
the entries along the (anti)diagonals. We start with (0,0) then
traverse (1,0) and (0,1) followed by (2,0), (1,1) and (0,2), and so on,
giving the sequence A054225:<br>
<br>
1 1 1 2 2 2 3 4 4 3 5 7 8 7 5 7 12 15 15 12 7 11 19 26 28 26 19 11 15 30
43 49...<br>
<br>
Note that sequencing the table has implicitly associated each pair (i,j)
with an index integer. For example, (0,0) is numbered 0, (1,1) is
numbered 4, etc, thus:<br>
<br>
T2 | 0
1 2
3 4
5 6
7 8 9 ...<br>
---+-------------------------------------------------------------<br>
0 | 0
2 5 9
14 20 27
35 44 54<br>
1 | 1
4 8 13
19 26 34
43 53 64<br>
2 | 3
7 12 18
25 33 42
52 63 75<br>
3 | 6
11 17 24
32 41 51
62 74 87<br>
4 | 10 16
23 31 40
50 61 73
86 100<br>
5 | 15 22
30 39 49
60 72 85
99 114<br>
6 | 21 29
38 48 59
71 84 98 113
129<br>
7 | 28 37
47 58 70
83 97 112 128
145<br>
8 | 36 46
57 69 82
96 111 127 144 162<br>
9 | 45 56
68 81 95 110
126 143 161 180<br>
<br>
<br>
<font size=3>Also note that there is a subtle but interesting difference
between the integers entered in table T1 and those in T2. The T1
entries actually COUNT something (namely pair partitions), hence they are
actual (“cardinal”) numbers. In contrast, the (“ordinal”) T2
integers really just NAME the corresponding pairs.<br>
<br>
To highlight this, we’ll use the term “numbral” (from “umbral numeral”)
when we use integers to “stand for” some other objects, as in T2 (as
opposed to normal numerical quantities, as in T1). When we want to
visually distinguish numbral integers from numeric integers we’ll write
the integer in brackets. We will also say that the object is “cast”
as the numbral, and conversely that the numbral “shadows” the
object.<br>
<br>
<br>
<font size=3>Thus the pair (0,0) is cast as [0], the numbral [4] shadows
(1,1), and generally, the “diagonal” numbering in T2 casts the pair (i,j)
as some numbral [n]:<br>
<br>
2<br>
(i+j) + i + 3j<br>
(i,j)
<--> [n] = [ -------------- ]<br>
2<br>
<br>
<font size=3>Related expressions to map from [n] back into the pair can
be computed based on, eg, rounding sqrt(2n+2). So we also have
sequences for the i-parts and j-parts:<br>
<br>
0 1 0 2 1 0 3 2 1 0 4 3 2 1 0 5 4 3 2 1 0 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 8
7 6...<br>
0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 0
1 2...<br>
<br>
<br>
Unsurprisingly these simple sequences are already in the OEIS, under
A025581 and A002262 respectively. We include them here in our
survey of the diagonal pair numbrals just to illustrate how we can
extract “component” sequences from a numbral sequence. For other
systems of numbrals this can yield novel results.<br>
<br>
Furthermore, given a shadow-cast relationship, we can proceed to define
various numbral operations as the images of operations on the
objects. For example the shadow (image) of pair addition, cast into
diagonal numbrals, specifies that<br>
<br>
[1] + [1] =
[3]<br>
<br>
(in this particular system, namely that of diagonal pair numbrals)
because<br>
<br>
(1,0) + (1,0)
= (2,0)<br>
<br>
and so on. The pair addition table cast in diagonal numbrals looks
like this:<br>
<br>
T3 | 0
1 2
3 4
5 6
7 8 9 ...<br>
---+-------------------------------------------------------------<br>
0 | 0
1 2
3 4
5 6
7 8 9<br>
1 | 1
3 4
6 7 8
10 11 12 13<br>
2 | 2
4 5
7 8 9
11 12 13 14<br>
3 | 3
6 7 10
11 12 15
16 17 18<br>
4 | 4
7 8 11
12 13 16
17 18 19<br>
5 | 5
8 9 12
13 14 17
18 19 20<br>
6 | 6
10 11 15
16 17 21
22 23 24<br>
7 | 7
11 12 16
17 18 22
23 24 25<br>
8 | 8
12 13 17
18 19 23
24 25 26<br>
9 | 9
13 14 18
19 20 24
25 26 27<br>
<br>
<font size=3>Note that in this table not only are the entries themselves
numbrals, but the rows and columns correspond to numbrals. We can
diagonally sequence this addition operation table as usual, giving the
sequence (of numbrals) A054237:<br>
<br>
<font size=3>0 1 1 2 3 2 3 4 4 3 4 6 5 6 4 5 7 7 7 7 5 6 8 8 10 8 8 6 7
10 9 11 11 9 10 7 8...<br>
<br>
<br>
This by no means exhausts what can be mined out of just the simple
diagonal pair numbrals. For instance we can count restricted
partitions, such as into distinct pairs (sequence A054242). This is
because any given additive numbral theory is typically at least as rich
as regular additive number theory. However we will move on to
introduce another numbral system...<br>
<br>
<br>
<font size=3>Often casting a given set of base objects into numbrals in
more than one way is a fecund source of many interesting derived
sequences. For instance an alternative to the diagonal sequencing
of pairs is to cast the pair (i,j) by interleaving the bits in the binary
expansion of i with those of j. For example (5,6) is thereby cast
into the bit-interleaved numbral [57] as follows:<br>
<br>
Bits of i= 5:
...0 0 1 0 1<br>
Bits of j= 6:
... 0 1 1 0<br>
-------------------------------------<br>
Bits of n=57:
...0 0 0 1 1 1 0 0 1<br>
<br>
This casting results in an alternative pair-naming table analogous to
T2:<br>
<br>
T4 | 0
1 2
3 4
5 6
7 8 9 ...<br>
---+-------------------------------------------------------------<br>
0 | 0
2 8 10
32 34 40
42 128 130<br>
1 | 1
3 9 11
33 35 41
43 129 131<br>
2 | 4
6 12 14
36 38 44
46 132 134<br>
3 | 5
7 13 15
37 39 45
47 133 135<br>
4 | 16 18
24 26 48
50 56 58 144
146<br>
5 | 17 19
25 27 49
51 57 59 145
147<br>
6 | 20 22
28 30 52
54 60 62 148
150<br>
7 | 21 23
29 31 53
55 61 63 149
151<br>
8 | 64 66
72 74 96
98 104 106 192 194<br>
9 | 65 67
73 75 97
99 105 107 193 195<br>
<br>
The recursively structured table T4 diagonalizes as the numbral sequence
A054238:<br>
<br>
0 1 2 4 3 8 5 6 9 10 16 7 12 11 32 17 18 13 14 33 34 20 19 24 15 36 35 40
21...<br>
<br>
Using T4 we can construct a bit-interleaved numbral addition table,
analogous to the construction of T3:<br>
<br>
T5 | 0
1 2
3 4
5 6
7 8 9 ...<br>
---+-------------------------------------------------------------<br>
0 | 0
1 2
3 4
5 6
7 8 9<br>
1 | 1
4 3
6 5 16
7 18 9
12<br>
2 | 2
3 8
9 6 7
12 13 10 11<br>
3 | 3
6 9 12
7 18 13
24 11 14<br>
4 | 4
5 6 7
16 17 18
19 12 13<br>
5 | 5
16 7 18
17 20 19
22 13 24<br>
6 | 6
7 12 13
18 19 24
25 14 15<br>
7 | 7
18 13 24
19 22 25
28 15 26<br>
8 | 8
9 10 11
12 13 14
15 32 33<br>
9 | 9
12 11 14
13 24 15
26 33 36<br>
<br>
The table T5 diagonalizes to give the sequence A054240:<br>
<br>
0 1 1 2 4 2 3 3 3 3 4 6 8 6 4 5 5 9 9 5 5 6 16 6 12 6 16 6 7 7 7 7 7 7 7
7 8...<br>
<br>
Since T2 just diagonalizes into the integers, T4 must be a permutation of
Z. The inverse of this permutation is the sequence A054239:<br>
<br>
<font size=3>0 1 2 4 3 6 7 11 5 8 9 13 12 17 18 24 10 15 16 22 21 28 29
37 23 30 31 39 38...<br>
<br>
These two sequences can be used to interconvert between diagonal pair and
bit-interleaved pair numbrals. For instance (coming full circle
with these examples) the number of partitions of bit-interleaved pair
numbrals is given by the sequence A054241:<br>
<br>
1 1 1 2 2 3 4 7 2 4 3 7 8 15 15 28 5 7 12 19 11 15 30 45 26 43 49 81 69
107...<br>
<br>
This sequence can be derived directly by using the addition table T5 to
construct the partitions. Equivalently, it can be generated by
reading out the elements of T1 in the order given by T4 (bit-interleaved)
rather than T2 (diagonal).<br>
<br>
<br>
That's probably enough for one message. In the next installment we
can develop aspects of the interleaved numbrals to derive
"base sqrt(2)", "Tinker Toy" and other, more exotic,
numbrals...<br>
<br>
</font><font size=3>Thanks!<br>
<br>
</font><br>
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