[seqfan] Re: 999999999888888887777777666666555554444333221

[Antti Karttunen]

On Mon, Sep 22, 2014 at 11:11 AM, Antti Karttunen <antti.karttunen at gmail.com
> wrote:

>
> > Message: 4
> > Date: Sat, 20 Sep 2014 19:11:14 -0400
> > From: Frank Adams-Watters <franktaw at netscape.net>
> > To: seqfan at list.seqfan.eu
> > Subject: [seqfan] Re: 999999999888888887777777666666555554444333221
> > Message-ID: <8D1A33F4F37D9C6-1720-11412 at webmail-m245.sysops.aol.com>
> > Content-Type: text/plain; charset="utf-8"; format=flowed
> >
> > There are 66712890763701234740813164553708284 terms in the sequence, if
> > I haven't made a mistake in my PARI program.
> >
> > a(b) = {my(r=vector(b*(b-1)\2+1));
> >    r[1] = 1;
> >    for(d=1,b-1,
> >       forstep(n=d*(d+1)\2,d,-1,
> >          r[n+1]+=binomial(n,d)*r[n-d+1]));
> >    r}
> > v = a(10)
> > sum(n=2,#v,v[n])
> >
> > By the number of digits, there are:
> >
> > 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17965, 44407, 201751, 801515,
> > 4890886, 52218595, 165519640, 835947970, 4290442728, 24096524166,
> > 179566203960, 2739764737710, 9938147178960, 60997160143920,
> > 331360222255920, 2154105076695000, 14308355062630200,
> > 148898652724750500, 3043362702904524000, 12550859255187653400,
> > 85564729840752162000, 446033694177751680000, 3160644316242901488000,
> > 23904928042959835872000, 212227787619709557696000,
> > 2872257514324824658032000, 85739562818913709978272000,
> > 359325740171513750386752000, 1764944667656072549494848000,
> > 10004773552120178696264400000, 69366429961366572294099840000,
> > 546927620849236435395787200000, 6125589353511448076432816640000,
> > 65850085550248066821652778880000, 1448701882105457470076361135360000,
> > 65191584694745586153436251091200000
> >
> > This is the vector returned by the function a above, except that that
> > vector has an additional leading 1. This is for zero-digit numbers;
> > i.e., zero if we wrote it as the empty string instead of the way we do.
> >
> > The number of values in bases 2 through 9 is:
> >
> > 1, 5, 80, 14381, 40885253, 2163451135829, 2525544441942679544,
> > 75742010013818779294524291
> >
> > This also is not in the OEIS.
> >
> > Franklin T. Adams-Watters
> >
>
> This theme lends itself naturally also to the factorial base,
> http://oeis.org/A007623
>
> In factorial base    In decimal
> (A007623)
>       1                1
>     221               17 (= 2*6 + 2*2 + 1)
>  233321             1895 (= 2*720 + 3*120 + 3*24 + 3*6 + 2*2 + 1)
>  323321             2495 (= 3*720 + 2*120 + 3*24 + 3*6 + 2*2 + 1)
>  332321             2591 (= 3*720 + 3*120 + 2*24 + 3*6 + 2*2 + 1)
>  333221
>
>
> 1, 17, 1895, 2495, 2591: "Sorry, but the terms do not match anything in
> the table."
>
> Also, how many terms there are whose width in factorial base is A000217(n)
> ? (Any digit d can at "earliest" occur as the d-th least significant digit).
>
> Cheers,
>
> Antti
>
>
Of course the sequence suggested above is also a subsequence of "Numbers n
whose factorial base representation A007623(n) doesn't contain any
nonleading zeros.":
http://oeis.org/A227157
to whose Crossrefs-lines I have collected various other subsequences, some
which have also a combinatorial role.

A.