# [seqfan] Digital convolution sequences

Wouter Meeussen eu000949 at pophost.eunet.be
Thu May 21 13:44:02 CEST 1998

```David,

>What would a Mathematica formula for this type of sequence look
>like?  Maybe someone should add one to these sequences so that they
>can be interpreted at a later date, as they have no published references
>(to my knowledge).
>
it can be :

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

Remark that the convolution can be defined as either
a[n-3]   a[n-2]  a[n-1]
*   ( u   ,   v,      w ) in base b
-----------------------
#   +   #   +   #   = a[n]

or just as well as

a[n-1]   a[n-2]  a[n-3]
*   ( u   ,   v,      w ) in base b
-----------------------
#   +   #   +   #   = a[n]

in this last case, you get (base 2):

Table[w[i,2],{i,0,50}]
=
{1,1,1,2,2,3,5,10,10,13,23,43,66,122,231,462,462,528,759,1452,1980,3201,
5412,9603,15015,26598,47025,88638,162261,312939,610863,1221726,1221726,
1310364,1623303,2547105,3768831,6300921,9234588,14715360,21016281,32797974,
53814255,90366045,144180300,245507889,422486163,743644737,1166130900,
2000141682,3411780471}

and in base 3 :

Table[w[i,3],{i,0,50}]
=
{1,1,2,2,4,8,16,40,112,112,152,376,528,1056,2336,4448,10176,23744,47488,
105152,257792,620736,1604416,4345152,11899136,34092992,100674560,100674560,
112573696,180759680,281434240,494682496,900589568,1463458048,2734257280,
5524801408,8259058688,15247318144,28974891392,52481268224,104962536448,
216913332352,426838405248,906158078720,1976748221568,3789064379008,
8169399227392,17559844142848,37687706976640,84765858868736,194039915419648}

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
>
>In this particular sequence, it is easy to see that after 6, every value
>must be even.  In general, it appears that for any given b and sufficient
>n, a(n) is divisible by every prime (possibly every number) strictly less
>than b (though the sufficient n can be very large indeed).  For primes p
>greater than or equal to b, it appears that there are arbitrarily large
>a(n) not divisible by p.
>
>For lack of a better name, I will call these "Digital convolution
>sequences."  For NJAS, I have computed them for bases b = 2 through
>10.  Note that as b increases, the sequences approach the factorial
>sequence (easy enough to prove).
>
>What would a Mathematica formula for this type of sequence look
>like?  Maybe someone should add one to these sequences so that they
>can be interpreted at a later date, as they have no published references
>(to my knowledge).
>
>%I A000002
>%S A000002
1,1,1,2,1,2,3,6,1,3,4,13,4,11,21,49,13,17,24,62,66,103,145,338,128,297,
>%T A000002
376,1156,763,1564,2592,6451,376,1532,1139,4235,4124,11714,8735,26105,
>%U A000002
5263,21212,18122,77153,35210,100649,135748,369972,95275,207638,231134
>%N A000002 Base-2 digital convolution sequence
>%A A000002 David Wilson (wilson at ctron.com)
>%O A000002 0,4
>%K A000002 base,nonn
>
>%I A000003
>%S A000003
1,1,2,1,3,7,6,20,52,6,26,104,32,162,460,356,1438,4048,712,3588,15272,
>%T A000003 5012,27460,90476,64944,300816,912472,90476,155420,611656,1067892,
>%U A000003 1770024,4763360,4151704,14746316,39566064,8915064,27813084,109938548
>%N A000003 Base-3 digital convolution sequence
>%A A000003 David Wilson (wilson at ctron.com)
>%O A000003 0,3
>%K A000003 base,nonn
>
>%I A000004
>%S A000004 1,1,2,6,2,8,18,62,36,160,392,1496,1176,5664,14856,61560,5664,20520,
>%T A000004
102600,313464,123120,539184,1514952,5207160,3569088,15498360,43342056,
>%U A000004 164591976,145524528,682642512,1966450584,8092803816,1365285024
>%N A000004 Base-4 digital convolution sequence
>%A A000004 David Wilson (wilson at ctron.com)
>%O A000004 0,3
>%K A000004 base,nonn
>
>%I A000005
>%S A000005 1,1,2,6,24,6,30,66,228,978,456,2412,5736,22032,99600,66096,364896,
>%T A000005 928080,3878928,18299952,15515712,88715520,239493888,1073343744,
>%U A000005 5251350528,239493888,1312837632,7877025792,23870571264,96795122688
>%N A000005 Base-5 digital convolution sequence
>%A A000005 David Wilson (wilson at ctron.com)
>%O A000005 0,3
>%K A000005 base,nonn
>
>%I A000006
>%S A000006 1,1,2,6,24,120,24,144,312,1080,4632,24240,9264,57744,134016,517536,
>%T A000006 2338176,12725952,7014528,45192384,111428352,469862208,2213733888,
>%U A000006 12478256064,8854935552,58767959808,152955661824,693938824704
>%N A000006 Base-6 digital convolution sequence
>%A A000006 David Wilson (wilson at ctron.com)
>%O A000006 0,3
>%K A000006 base,nonn
>
>%I A000007
>%S A000007
1,1,2,6,24,120,720,120,840,1800,6240,26760,140040,867000,280080,2014080,
>%T A000007
4588320,17793120,80349120,437331840,2784689280,1311995520,9666063360,
>%U A000007 23268113280,98802529920,465014459520,2621479887360,17123922702720
>%N A000007 Base-7 digital convolution sequence
>%A A000007 David Wilson (wilson at ctron.com)
>%O A000007 0,3
>%K A000007 base,nonn
>
>%I A000008
>%S A000008 1,1,2,6,24,120,720,5040,720,5760,12240,42480,182160,953280,5901840,
>%T A000008
42266160,11803680,96336000,216279360,841510080,3798599040,20676015360,
>%U A000008 131653290240,962925062400,394959870720,3283735057920,7752349728000
>%N A000008 Base-8 digital convolution sequence
>%A A000008 David Wilson (wilson at ctron.com)
>%O A000008 0,3
>%K A000008 base,nonn
>
>%I A000009
>%S A000009
1,1,2,6,24,120,720,5040,40320,5040,45360,95760,332640,1426320,7464240,
>%T A000009 46211760,330946560,2693784240,661893120,6049461600,13422709440,
>%U A000009 52367051520,236313624960,1286302227840,8190440616960,59905688774400
>%N A000009 Base-9 digital convolution sequence
>%A A000009 David Wilson (wilson at ctron.com)
>%O A000009 0,3
>%K A000009 base,nonn
>
>%I A000010
>%S A000010 1,1,2,6,24,120,720,5040,40320,362880,40320,403200,846720,2943360,
>%T A000010 12620160,66044160,408885120,2928240000,23834805120,217441486080,
>%U A000010 47669610240,482552582400,1060444385280,4146438320640,18706642053120
>%N A000010 Base-10 digital convolution sequence
>%A A000010 David Wilson (wilson at ctron.com)
>%O A000010 0,3
>%K A000010 base,nonn
>
>
>
>
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be

```