[seqfan] Re: 999999999888888887777777666666555554444333221

[Frank Adams-Watters]

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

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>

(My last post of the day, Gérard!)

Hello SeqFans,
Numbers showing, for any of their digit "d", exactly d copies of "d":

S=1,22,122,212,221,1333,3133,3313,3331,14444,41444,44144,44414,44441,...

Example:
312332 belongs to S as there are exactly
one "1", two "2" and three "3" in 312332.

The sequence is finite (how many
terms?) and the last term is, of course,
this message header.

Best,
É.


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