[seqfan] Re: Comma numbers (2)

[Farideh Firoozbakht]

Hello Eric,

>> Let a(1) = 0
>>     a(2) = 1
>> and a(n) = a(n-1) + [the two-digit integer split by the comma
>                      which separates a(n-1) and a(n-2)]

>> S = 0,1,1,12,23,45,78,135,216,268,330,413,417,451,525,540,595,600,
>>     656,662,728,755,...

> ... no, 78 should be 79 :-(

  According to the definition it seems a(3) should be 2, because
  a(3) = a(2) + 01 = 1 + 1 = 2 .

  The first 100 terms of the sequence :

  0, 1, 2, 14, 35, 78, 135, 216, 268, 330, 413, 417, 451, 525, 540,
  595, 600, 656, 662, 728, 755, 842, 900, 929, 938, 1037, 1118, 1189,
  1270, 1361, 1362, 1373, 1394, 1425, 1466, 1517, 1578, 1649, 1730,
  1821, 1822, 1833, 1854, 1885, 1926, 1977, 2038, 2110, 2192, 2194,
  2216, 2258, 2320, 2402, 2404, 2426, 2468, 2530, 2612, 2614, 2636,
  2678, 2740, 2822, 2824, 2846, 2888, 2950, 3032, 3035, 3058, 3111,
  3194, 3207, 3250, 3323, 3326, 3359, 3422, 3515, 3538, 3591, 3674,
  3687, 3730, 3803, 3806, 3839, 3902, 3995, 4018, 4072, 4156, 4180,
  4244, 4248, 4292, 4376, 4400, 4464


  The Mma coding for this sequence :

  a[1]=0;a[2]=1;
  a[n_]:=a[n]=a[n-1]+10 Mod[a[n-2],10]+IntegerDigits[a[n-1]][[1]];
  Table[a[k],{k,100}]


  --- Farideh


Quoting Eric Angelini <Eric.Angelini at kntv.be>:

>
> Aaaaarghllll :
>
>> 78 is in S because we add to 45 the integer 34 seen around the   
>> comma of [23,45]
>
> ... no, 78 should be 79 :-(
>
> I don't know now about "convergence" any more... I guess it might   
> still happen...
>
> You've got the general idea anyway
> Sorry for my slow brain and computations
> Best,
> ?.
>
>
> -----Message d'origine-----
> De : seqfan-bounces at list.seqfan.eu   
> [mailto:seqfan-bounces at list.seqfan.eu] De la part de Eric Angelini
> Envoyé : jeudi 15 octobre 2009 15:20
> ? : Sequence Fanatics Discussion list
> Objet : [seqfan] Comma numbers
>
>
> Hello SeqFans,
>
> Let a(1) = 0
>     a(2) = 1
> and a(n) = a(n-1) + [the two-digit integer split by the comma
>                      which separates a(n-1) and a(n-2)]
>
> S =   
> 0,1,1,12,23,45,78,135,216,268,330,413,417,451,525,540,595,600,656,662,728,755,...
>
> 23 is in S because we add to 12 the integer 11 seen around the comma  
>  of [1,12]
> 78 is in S because we add to 45 the integer 34 seen around the comma  
>  of [23,45]
> 417 is in S because we add to 413 the integer 4 seen around the   
> comma of [330,413]
>
> Note(1):
>
> Those two different starts converge towards the same seq:
>
> Sa = 4,16,57,122,193,214,246,288,350, ...
> Sb = 4, 17, 58, 133, 214,246,288,350, ...
>
> What are the laws ruling convergence?
>
> Note(2):
>
> I am looking for "comma numbers", which are numbers like [abc] where  
>  the split [a,bc]
> or [ab,c] would produce later in the sequence the said "comma   
> number" again: ..., abc, ...
>
> Example: 416 is not a "comma number" because we have no hit for the   
> two different
>          starts [4,16] or [41,6]:
>
> S1 = 4,16,57,122,193,214,246,288,350,433, ... <-- no hit
> S2 = 41,6,22,84,112,153,174,205,247,299,371,464, ... <-- no hit
>
> I guess 10 is the first "comma number":
>
> 10 --> 1,0,10,11,12,23, etc. --> '10' is in the seq
> (we see that '20', '30', '40', etc. are in the seq too)
>
> What about 11?
>
> 11 --> 1,1,12,23, etc. <-- no hit: '11' is not a "comma number"
>
> How would the "comma numbers" seq look like?
>
> Best,
> ?.
>
> (this was inspired by:
>
> A121805 The "commas" sequence: a(1) = 1; for n > 1, let x be the   
> least significant digit of a(n-1); then a(n) = a(n-1) + x*10 + y   
> where y is the most significant digit of a(n) and is the smallest   
> such y, if such a y exists. If no such y exists, stop.
>

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>








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