Yet about tables.

Antti Karttunen karttu at
Sun Jul 25 23:38:22 CEST 1999

N. J. A. Sloane wrote:
> Antti, there are not so many sequences with
> irregular shapes.  and i'm very busy with other
> projects (and my summer student has just about
> finished).
> so i'm not going to follow the suggestions in your
> proposal!   but i will keep them on record for
> the future.

Neil, I'm sorry for the insipidity, it was just my (very 
selective) perfectionism that drove for that.
And yes, indeed, by searching for a word "tabf"
one can currently find only the following

%K A012257 nonn,tabf,nice
%K A027113 nonn,tabf
%K A036038 nonn,easy,nice,tabf
%K A028297 tabf,easy,sign,done
%K A027926 nonn,tabf
%K A046816 nonn,tabf,easy
%K A046752 nonn,nice,tabf,more
%K A047971 nonn,easy,nice,tabf
%K A036039 nonn,easy,nice,tabf
%K A008855 nonn,tabf,nice
%K A036040 nonn,easy,nice,tabf

Of these, only A028297 is of the format 1,1,2,2,3,3,4,4,
... terms, and an alternative permutation of the sequence
with 1,2,3,4,5,etc. terms could be easily produced.
And only A027113 and A027926 seem to be of the format
1,3,5,7,9,... All the others are of various other,
even funnier formats. 

Especially nice is A046816 (Entries in 3-dimensional version 
of Pascal triangle: trinomial coefficients)
a la:
           ... 1 .... Here is the third slice of the pyramid
           .. 3 3
           . 3 6 3
           .1 3 3 1
so the next project should be of course a 3D VRML showscript 
for the 3D-recurrences like this. Fortunately, I'm not going 
to do it.

> all the best
> Neil

>And Michael wrote:

First about the general problems of indexing.
My comment: Would not some of the sequences make actually
more sense as sets than as sequences with definite indexing?
I.e. ordinary primes is a set of irreducible integers
(with certain property that defines them, but not any
actual recurrence or formula that would tie them together)
and we are just used to seeing them in the order of 
magnitude, 2,3,5,7,11,... (well, which is of course the
side-product of the Sieve of Erasthotenes).
Then consider semi-primes:
A001358 (Formerly M3274 and N1323)
Sequence: 4,6,9,10,14,15,21,22,25
but as well we could list them in order
4,6,9,10,15,25,21,35,49 which would suddenly make
very much sense when read as an upper triangular array:

4 6 10 14
  9 15 21
    25 35

>Here is a simple example:

>        1   2   4   7  11 ...
>        3   5   8  12 ...
>        6   9  13 ...
>       10  14 ...
>       15 ...
>       ...
>This is the positive integer sequence arranged in a 
>triangular arrangement which zig-zags along the 
>anti-diagonals of a square array. 
>We even have simple variations like reading by the 
>going up instead of down giving [1,3,2,6,5,4,10,9,8,7,...].

That does not matter with the current script. What the user
will see is just a reflection of what he would see if
the sequence's author had submitted it "correctly", so we get
 1 3 6 10
 2 5 9
 4 8
from your example reading.

Of course: if somebody submits "boustrophedon antidiagonal" 
(real zig-zag) readings of tables into the Encyclopedia, it's 
best to regard them as "boustrophedon antidiagonal 
permutations" of the original sequence, and not to try render 
the sequence straight again.

>If the sequence is given by a natural description, for 
>a(n,k) = number of k-subsets of an n-set, then this is 
>important information. This is more important than whether it is displayed

>          1               1   1   1   1         1
>        1   1      or     1   2   3   4   or    1  1
>      1   2   1           1   3   6  10         1  2  1
>    1   3   3   1         1   4  10  20         1  3  3  1

I wholly agree. The whole point of showtable script was
just to give some kind of a visual clue what's going on,
as well as to help spotting any additional patterns
on any particular row or column. And it might be helpful
also with sequences not usually even thought as "2D arrays". 
I.e. all the sequences computed of two somethings (especially 
with sums, and see my example about semi-primes above).

And furthermore, all the current three formats
(as well as the third one in the example above)
are in a way just reflections/twistortions of one and single 
triangle/table format, one where terms are arranged in groups 
of 1,2,3,4,etc, and the choice among these three formats
is just visual sugaring.

Now, some sequences really have ambiguous indexing.

Name:      Exponent of highest power of 2 dividing n (the 
binary carry sequence)

This has been stored with the offset 1.
However, as the column/row 1 of table A050602 it makes
sense only as starting from the offset 0.
The difference: with the offset 1 it tells the number
of the trailing zeroes in the binary expansion of n, while 
with the offset 0 the number is that of the trailing ones.


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