[seqfan] Re: a(n) = a(n-1) + [sum of all digits present so far]

Eric Angelini Eric.Angelini at kntv.be
Wed Oct 7 11:59:40 CEST 2009


Thanks, Farideh !
Best,
É.
 

-----Message d'origine-----
De : seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu] De la part de Farideh Firoozbakht
Envoyé : jeudi 1 octobre 2009 20:32
À : Eric Angelini; mymontain at yahoo.com
Cc : Sequence Fanatics Discussion list
Objet : [seqfan] Re: a(n) = a(n-1) + [sum of all digits present so far]


  Hello Eric,

  > S = 1, 2, 5, 13, 25, 44, 71, 106, 113, ...
  > a(1) = 1
  > a(n) = a(n-1) + [sum of all digits present so far in S]

  According to the definition of a(n) we have the following recursion
  formula for a(n):

  a(1)=1,a(2)=2; a(n+1)=2a(n)-a(n-1)+sod(a(n))

  The correct value of a(9) is 148 , you wrote 113.

  a(9)=106+1+2+5+(1+3)+(2+5)+(4+4)+(7+1)+(1+0+6)=148

  or a(9)=2a(8)-a(7)+sod(a(8))=2*106-71+(1+0+6)=148

  The first 100 terms of this sequence:

  1, 2, 5, 13, 25, 44, 71, 106, 148, 203, 263, 334, 415, 506, 608, 724,
  853, 998, 1169, 1357, 1561, 1778, 2018, 2269, 2539, 2828, 3137, 3460,
  3796, 4157, 4535, 4930, 5341, 5765, 6212, 6670, 7147, 7643, 8159, 8698,
  9268, 9863, 10484, 11122, 11767, 12434, 13115, 13807, 14518, 15248,
  15998, 16780, 17584, 18413, 19259, 20131, 21010, 21893, 22799, 23734,
  24688, 25670, 26672, 27697, 28753, 29834, 30941, 32065, 33205, 34358,
  35534, 36730, 37945, 39188, 40460, 41746, 43054, 44378, 45728, 47104,
  48496, 49919, 51374, 52849, 54352, 55874, 57425, 58999, 60613, 62243,
  63890, 65563, 67261, 68981, 70733, 72505, 74296, 76115, 77954


  The Mathematica coding for this sequence:

  a[1]=1;a[2]=2;a[n_]:=a[n]=2*a[n-1]-a[n-2]+Apply[Plus,IntegerDigits[a[
  n-1]]];Table[a[n],{n,100}]


  Best regards,
  Farideh



Quoting Jonathan Post <jvospost3 at gmail.com>:

> It's an infinite array, indexed by the initial value(s).  I suggest
> showing the array explicitly (at least a 2-D slice) and the main
> diagonal as a derived sequence.
>
> -- Prof. Jonathan Vos Post
>
> On Thu, Oct 1, 2009 at 9:23 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
>>
>> Hello SeqFans,
>> this seems not to be in the OEIS (else I need new glasses):
>>
>> S = 1, 2, 5, 13, 25, 44, 71, 106, 113, ...
>>
>> a(1) = 1
>> a(n) = a(n-1) + [sum of all digits present so far in S]
>>
>> (thus 25 = 13 + 1 + 2 + 5 + 1 + 3)
>>
>> Best,
>> É.
>>
>>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>








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