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<TITLE>Re: [math-fun] Sum of last ten digits</TITLE>
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Hi,<BR>
Concerning the loopings of sequences like:<BR>
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<BR>
%S A112395 0,0,0,0,0,0,0,0,0,1,1,2,4,8,16,23,28,37,44,40,37,42,38,39,43,46,46,50,<BR>
%N A112395 Next term is the sum of the last 10 digits in the sequence.<BR>
%C A112395 There are only 10^10 possibilities for the last 10 digits, so the sequence must eventually cycle.<BR>
%C A112395 Terms computed by Gilles Sadowski.<BR>
%e A112395 0 + 0 + 1 + 1 + 2 + 4 + 8 + 1 + 6 = 23<BR>
%A A112395 Eric Angelini (eric.angelini(AT)kntv.be), Dec 05 2005<BR>
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-----------<FONT COLOR="#800000">Hans:<BR>
In fact, terms 19-23 (44, 40, 37, 42, 38) are repeated by terms 331-335 already.<BR>
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-----------<FONT COLOR="#000080">Alexandre:<BR>
It seems the shortest loop of Éric Angelini's < sum of last ten digits > can be found beginning with 9, and has a lenght of 1 term:<BR>
0 0 0 0 0 0 0 0 0 9 9 18 27 36 45 54 45 45 45 45 45 45 45 45 45 <BR>
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-----------<FONT COLOR="#0000FD">Graeme<BR>
Searching randomly, I found that:<BR>
A loop of length 8 is possible, starting at 0, 6, 1, 8, 7, 8, 6, 6, 3, 0.<BR>
A loop of length 24 is possible, starting at 0, 4, 2, 1, 9, 7, 1, 7, 7, 4.<BR>
A loop of length 26 is possible, starting at 2, 3, 2, 7, 0, 9, 8, 7, 8, 4.<BR>
A loop of length 78 is possible, starting at 2, 2, 5, 2, 6, 0, 3, 8, 5, 7.<BR>
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-----------<FONT COLOR="#008080">Hans <BR>
For example, starting with '0,0,0,0,0,0,0,0,1' and letting the <BR>
sequence reflect the sum of the last '9 digits' (instead of 10) the <BR>
loop-size is 12203 (starting with term 14250), in stark contrast to <BR>
A112402's loop-length of only 312.<BR>
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<BR>
I asked to Michael D Beeler if he could do the same job (as he did before in different bases, all beginning with 0,0,...0, 1), but here starting with s-1 zeros followed by a 2, a 3, ...a 9?<BR>
Here is his complete answer in base 10. Some patterns...<BR>
And sequences style "Lenght of loop for s=n beginning with (s-1) zero's followed by the digit i"<BR>
<BR>
Best.<BR>
Alexandre<BR>
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------ Message transféré<BR>
<BR>
Yes, it is not hard to get the cases you ask about.<BR>
The results are below. I am assuming you are interested<BR>
only in decimal (radix = base = b = 10). In the table<BR>
below, "digit" is the first non-zero digit. So digit=1<BR>
is identical to the original problem.<BR>
<BR>
s=2 s=3 s=4 s=5 s=6 s=7 s=8 s=9 s=10<BR>
digit=1 8 4 50 171 14 461 78 12203 312<BR>
digit=2 8 10 50 171 39 461 78 12203 312<BR>
digit=3 8 10 12 171 13 461 26 12203 104<BR>
digit=4 8 4 50 171 39 461 78 12203 312<BR>
digit=5 8 4 50 63 39 461 78 12203 312<BR>
digit=6 8 10 12 171 69 461 26 12203 104<BR>
digit=7 3 15 50 171 69 461 78 12203 312<BR>
digit=8 8 4 50 171 39 461 78 12203 312<BR>
digit=9 8 10 1 171 1 461 1 12203 1<BR>
<BR>
s=11 s=12 s=13 s=14 s=15 s=16<BR>
digit=1 318880 2184 57725 5804 1401 9722<BR>
digit=2 318880 2184 57725 5804 1401 3251<BR>
digit=3 318880 728 57725 5804 6698 9722<BR>
digit=4 318880 2184 57725 5804 1401 3251<BR>
digit=5 318880 2184 57725 5804 6698 9722<BR>
digit=6 318880 728 57725 5804 6698 9722<BR>
digit=7 318880 2184 57725 5804 6698 5091<BR>
digit=8 318880 2184 57725 5804 1401 9722<BR>
digit=9 318880 1 57725 1 6698 1<BR>
<BR>
For example, with s=3 and digit=7, the sequence begins<BR>
0 0 7 and falls into a loop that is 15 terms long.<BR>
<BR>
Of course, just because one loop has the same number<BR>
of terms as another, that does not mean they are the<BR>
same loop. (0 0 0 7) and (0 0 0 8) each fall into<BR>
loops of 50 terms, but they might be different loops.<BR>
I did not investigate whether any of the same loop-lengths<BR>
in the table above are actually the same loop.<BR>
<BR>
There certainly are some curious patterns in the table.<BR>
<BR>
In number sequences like this, I think decimal is kind of<BR>
an arbitrary radix. That's why in my previous message I<BR>
generalized to various radixes. But computing a table<BR>
like that above for several different radixes is a bit<BR>
of work, and might be overwhelming in the volume of data.<BR>
Maybe it is nice to stay with decimal when varying the<BR>
first non-zero digit.<BR>
<BR>
Regards,<BR>
-- Mike<BR>
<BR>
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