numbral sequences
Marc LeBrun
mlb at well.com
Wed Feb 9 08:18:14 CET 2000
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...
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].
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:
(i,j) + (k,l) = (i+k,j+l).
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):
(2,1) (2,0)+(0,1) (1,0)+(1,0)+(0,1) (1,0)+(1,1)
We can arrange these pair partition counts in an (infinite) numeric table, T1:
T1 | 0 1 2 3 4 5 6 7 8 9 ...
---+-------------------------------------------------------------
0 | 1 1 2 3 5 7 11 15 22 30
1 | 1 2 4 7 12 19 30 45 67 97
2 | 2 4 8 15 26 43 69 107 163 242
3 | 3 7 15 28 49 81 130 203 311 466
4 | 5 12 26 49 86 143 231 362 558 841
5 | 7 19 43 81 143 236 381 595 916 1380
6 | 11 30 69 130 231 381 616 961 1480 2230
7 | 15 45 107 203 362 595 961 1492 2290 3440
8 | 22 67 163 311 558 916 1480 2290 3506 5251
9 | 30 97 242 466 841 1380 2230 3440 5251 7838
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).
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:
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...
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:
T2 | 0 1 2 3 4 5 6 7 8 9 ...
---+-------------------------------------------------------------
0 | 0 2 5 9 14 20 27 35 44 54
1 | 1 4 8 13 19 26 34 43 53 64
2 | 3 7 12 18 25 33 42 52 63 75
3 | 6 11 17 24 32 41 51 62 74 87
4 | 10 16 23 31 40 50 61 73 86 100
5 | 15 22 30 39 49 60 72 85 99 114
6 | 21 29 38 48 59 71 84 98 113 129
7 | 28 37 47 58 70 83 97 112 128 145
8 | 36 46 57 69 82 96 111 127 144 162
9 | 45 56 68 81 95 110 126 143 161 180
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.
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.
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]:
2
(i+j) + i + 3j
(i,j) <--> [n] = [ -------------- ]
2
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:
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...
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...
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.
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
[1] + [1] = [3]
(in this particular system, namely that of diagonal pair numbrals) because
(1,0) + (1,0) = (2,0)
and so on. The pair addition table cast in diagonal numbrals looks like this:
T3 | 0 1 2 3 4 5 6 7 8 9 ...
---+-------------------------------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 3 4 6 7 8 10 11 12 13
2 | 2 4 5 7 8 9 11 12 13 14
3 | 3 6 7 10 11 12 15 16 17 18
4 | 4 7 8 11 12 13 16 17 18 19
5 | 5 8 9 12 13 14 17 18 19 20
6 | 6 10 11 15 16 17 21 22 23 24
7 | 7 11 12 16 17 18 22 23 24 25
8 | 8 12 13 17 18 19 23 24 25 26
9 | 9 13 14 18 19 20 24 25 26 27
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:
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...
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...
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:
Bits of i= 5: ...0 0 1 0 1
Bits of j= 6: ... 0 1 1 0
-------------------------------------
Bits of n=57: ...0 0 0 1 1 1 0 0 1
This casting results in an alternative pair-naming table analogous to T2:
T4 | 0 1 2 3 4 5 6 7 8 9 ...
---+-------------------------------------------------------------
0 | 0 2 8 10 32 34 40 42 128 130
1 | 1 3 9 11 33 35 41 43 129 131
2 | 4 6 12 14 36 38 44 46 132 134
3 | 5 7 13 15 37 39 45 47 133 135
4 | 16 18 24 26 48 50 56 58 144 146
5 | 17 19 25 27 49 51 57 59 145 147
6 | 20 22 28 30 52 54 60 62 148 150
7 | 21 23 29 31 53 55 61 63 149 151
8 | 64 66 72 74 96 98 104 106 192 194
9 | 65 67 73 75 97 99 105 107 193 195
The recursively structured table T4 diagonalizes as the numbral sequence
A054238:
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...
Using T4 we can construct a bit-interleaved numbral addition table, analogous
to the construction of T3:
T5 | 0 1 2 3 4 5 6 7 8 9 ...
---+-------------------------------------------------------------
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 4 3 6 5 16 7 18 9 12
2 | 2 3 8 9 6 7 12 13 10 11
3 | 3 6 9 12 7 18 13 24 11 14
4 | 4 5 6 7 16 17 18 19 12 13
5 | 5 16 7 18 17 20 19 22 13 24
6 | 6 7 12 13 18 19 24 25 14 15
7 | 7 18 13 24 19 22 25 28 15 26
8 | 8 9 10 11 12 13 14 15 32 33
9 | 9 12 11 14 13 24 15 26 33 36
The table T5 diagonalizes to give the sequence A054240:
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...
Since T2 just diagonalizes into the integers, T4 must be a permutation of Z.
The inverse of this permutation is the sequence A054239:
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...
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:
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...
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).
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...
Thanks!
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