A108571: Total number of terms

[Antti Karttunen]

zak seidov wrote:

>Dear seqfans,
>
>Anybody may wish to check these results:
>
>Mumber of n-digit terms in A108571: (n=1..45)
>
>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.
>Summing this, total number of terms in A108571 is: 
>66712890763701234740813164553708284.
>Thanks, Zak
>
>  
>

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This is actually nice:

%I A108571
%S A108571 1,22,122,212,221,333,1333,3133,3313,3331,4444,14444,22333,23233,23323,
%T A108571 23332,32233,32323,32332,33223,33232,33322,41444,44144,44414,44441,
%U A108571 55555,122333,123233,123323,123332,132233,132323,132332,133223,133232
%N A108571 Any digit d in the sequence says: "I am part of an integer in which you'll find d digits "d".
%C A108571 The sequence is finite. Last term: 999999999888888887777777666666555554444333221.
%Y A108571 Sequence in context: A085828 A081931 A043498 this_sequence A105776 A044354 A044735
%Y A108571 Adjacent sequences: A108568 A108569 A108570 this_sequence A108572 A108573 A108574
%K A108571 base,easy,fini,nonn
%O A108571 1,2
%A A108571 Eric Angelini (eric.angelini(AT)kntv.be), Jul 05 2005

If one considers the same process in factorial base http://www.research.att.com/~njas/sequences/A007623
then (a similar) sequence can be made infinite. But then only few of these terms
can be used, because there is an additional condition that the nth digit from
the right can be at most n. So only 1, 221, 233321, 323321, 332321, 333221, etc. remain.
and the number of digits must be a triangular number.

Alternatively, one could consider vectors of arbitrarily large integers,
not decimal numbers with max. digit 9, and this "sequence of vectors" would begin
just like A108571, and the "number of n-element vectors matching the criteria"
would begin like your sequence. Actually, I think it should be this:
http://www.research.att.com/~njas/sequences/A007837
"Number of partitions of n-set with distinct block sizes."
which begins as
1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419,
i.e. differs from your sequence at a(11)=44419, which is not a coincidence. .-)

Cheers,
and Happy Beginning of the New Year,

Antti