[math-fun] Sum of last ten digits

[Graeme McRae]

Re :  [math-fun] Sum of last ten digitsSearching randomly, I found that:
A loop of length 8 is possible, starting at 0, 6, 1, 8, 7, 8, 6, 6, 3, 0.
A loop of length 24 is possible, starting at 0, 4, 2, 1, 9, 7, 1, 7, 7, 4.
A loop of length 26 is possible, starting at 2, 3, 2, 7, 0, 9, 8, 7, 8, 4.
A loop of length 78 is possible, starting at 2, 2, 5, 2, 6, 0, 3, 8, 5, 7.
--Graeme
  ----- Original Message ----- 
  From: Alexandre Wajnberg 
  To: Hans Havermann ; Eric ANGELINI ; N. J. A. Sloane ; Seqfan 
  Cc: Gilles ; Michael D Beeler 
  Sent: Thursday, December 08, 2005 4:37 AM
  Subject: Re : [math-fun] Sum of last ten digits



  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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