# [seqfan] Another few hundred thousand sequences?

Richard Guy rkg at cpsc.ucalgary.ca
Sun Nov 25 20:48:01 CET 2012

```Dear all,
I'm not sure to what extent number walls appear in
OEIS, but I suspect that it's not very great.  I've forgotten
how they originated, and they're not very widely known, but
you can read about them in Conway & Guy, The Book of Numbers.

Much better known are difference tables and I recently
mentioned  A000127, with its first differences  A000125, second
differences  A000124, and third differences  A000027, and it's
well known how one can use the Newton-Lagrange formula to obtain
a formula for the original sequence.

Similarly, one can use the Berlekamp-Massey algorithm
(see Lidl & Niederreiter) to retrieve the formula for a sequence
from its number wall:

1  1 1 1 1  1  1  1   1    1    1    1    1    1     1     1
1  1 2 4 8 16 31  57  99  163  256  386  562  794   1093 1471
-1 0 0 0  8 49 180 510 1225 2618 5125 9360 16170 26675
0 0 0  4 31 130 400 1015 2254 4536 8460 14850
0 0  2  9  25  55  105  182  294  450
1  1  1  1   1    1    1    1    1
0  0  0   0    0    0    0

Write down any integer sequence.  Surmount it by a row of ones.
Subsequent rows are obtained by the rule  N*S + W*E = centre^2.

``Windows'' of zeroes are a bit tricky to circumnavigate (see
C & G for details).

For example, the entry  130  above is obtained from

(57 * x)  +  (49 * 510)  =  180^2

The miracle is that no matter what integer sequence you start
with, subsequent rows are always integers.  But even if you
start from a reasonably simple sequence, eventually yeilding
a recognizable pattern, the intermediate rows are comparatively
complicated.  Here, for example, starting from  A000127, it's
not until the third row that we find an OEIS sequence, A005582,
interesting in its own right.

The formulas here are, writing  ``(n;r)''  for  ``n choose r''

Original sequence:  A000127   (n;4) + (n;2) + (n;0)
next row:  5(n;6) + 10(n;5) + 7(n;4) + (n;3) - (n;2) + (n;1) - (n;0)
next:      5(n;6) + 15(n;5) + 15(n;4) + 4(n;3)
next:  A005582      (n;4) + 3(n;3) + 2(n;2)

R.

```