[math-fun] Sum of last ten digits

[Alexandre Wajnberg]

Hi,

Neil said:
  %S A112402 0,0,0,0,0,0,0,0,0,1,1,2,4
> %N A112402 Next term is the sum of the last 10 digits in the sequence.
> %C A112402 Digits, not terms!
> %C A112402 There are only 10^10 possibilities for the last 10 digits, so the
> sequence must eventually cycle.
Hans:
In fact, terms 19-23 (44, 40, 37, 42, 38) are repeated by terms 331-335
already. 
Me:
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:
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
It should be interesting to taste the eight other (beginning with 2, 3 ...8)

Thanks to Gilles Sadowski's computing, the seven other loops are, beginning
with:

1: loop at n(19)=44, 312 terms (Hans)
2: loop at n(17)=38, 312 terms
3: loop at n(21)=36, 104 terms
4: loop at n(16)=38, 312 terms
5: loop at n(28)=42, 312 terms
6: loop at n(28)=36, 104 terms
7: loop at n(18)=44, 312 terms
8: loop at n(32)=37, 312 terms
9: loop at n(17)=45, 1   term
(with offset 1,)

I'm submitting them.
I don't know if this is of interest:
-only loops by lenght of 104 and 312 (and 1);
-a pattern: [312 312 104];
-loop 4 == loop 2; loop 6 = loop 3 but beginning at position 47 of loop 3;

What's happening beginning with 10, 11, 12... ?


Alexandre

Exemple:
2: loop at n(17)=38, 312 terms
0 0 0 0 0 0 0 0 0 2 2 4 8 16 23 28
38 45 42 41 41 36 34 32 31 30 28 29 33 34 37 44 42 37 41 39 41 38 43 40 39
39 46 45 47 54 51 45 44 43 39 42 42 39 43 43 38 43 44 40 37 40 33 32 29 36
35 39 45 49 51 48 52 47 49 49 56 55 58 60 53 48 49 52 46 50 47 46 43 43 40
39 40 34 34 34 37 35 39 44 45 47 48 52 47 50 46 45 42 41 35 38 39 42 42 43
42 37 35 37 41 39 45 44 44 42 43 38 40 36 37 41 39 40 40 35 33 34 29 36 41
38 43 43 39 42 43 39 44 45 42 42 41 34 33 30 27 30 28 31 29 37 38 46 46 52
48 50 44 42 38 42 36 40 36 39 40 38 40 40 35 31 31 24 26 30 25 28 34 35 35
40 37 37 40 36 37 43 40 34 37 38 39 44 48 53 51 46 44 44 40 36 39 41 38 41
42 39 39 46 45 49 56 55 53 51 48 47 47 48 52 53 49 51 46 44 45 46 43 44 42
40 35 33 32 29 34 37 39 45 49 51 50 45 42 39 38 43 45 45 48 48 49 55 56 58
59 61 55 55 54 50 41 39 41 36 36 40 39 39 46 47 49 58 59 61 58 60 53 48 46
49 49 56 59 61 58 58 58 60 52 52 46 43 37 41 39 44 42 41 36 40 32 29 34 36
36 41 41 35 36 36 36 40 39 43 41 37
38 45 42 41 41 36 
 







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