[seqfan] Digital convolution sequences

Wouter Meeussen eu000949 at pophost.eunet.be
Thu May 21 14:28:27 CEST 1998

```Dave,

if you choose the smallest elements of the sequence for convolution,
and not the biggest ones as you suggested,
then you get a sequence that rises very slowly :

con[0,_Integer]:=1;
con[n_Integer,b_Integer]:=con[n,b]=Module[{mul},
mul=IntegerDigits[n,b];
Table[con[i,b],{i,0,Length[mul]-1} ]. mul]

Table[con[i,2],{i,0,100}]
=
{1,1,1,2,1,2,2,3,1,3,2,4,2,4,3,5,1,2,3,4,2,3,4,5,2,3,4,5,3,4,5,6,1,3,2,4,3,5,
4,6,2,4,3,5,4,6,5,7,2,4,3,5,4,6,5,7,3,5,4,6,5,7,6,8,1,3,3,5,2,4,4,6,3,5,5,7,
4,6,6,8,2,4,4,6,3,5,5,7,4,6,6,8,5,7,7,9,2,4,4,6,3}

...make it run to 10,000 and get a nice Fourier transform of something
mildly chaotic.

too bad it doesn't *count* anything.

wouter.

At 13:07 19-05-98 -0400, David W. Wilson wrote:
>Here is an interesting idea for a class of integer sequences that I came
>up with some time ago but never pursued.
>
>Choose a base b >= 2.  Let a(0) = 1.  To compute subsequent a(n), write
>a(n) in base b, and convolve the digits with the immediately previous
>values of a.
>
>For example, if we choose b = 3, the values of a(0) through a(13)
>happen to be
>
>    1 1 2 1 3 7 6 20 52 6 26 104 32 162
>
>To compute a(14), we note that in base 3, 14 = 112, so we convolve
>the last three elements with 1, 1, 2 to obtain
>
>    1 1 2 1 3 7 6 20 52 6 26 104 32 162
>                              *1 *1  *2
>                             ----------
>                             104+32+324 = 460
>
*** Clip ***

>
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be

```