[seqfan] Re: Sum of a(n-1) and a(n)'s first digit

[David Wilson]

For n >= 0, Let S[n] be the sequence starting with n, and with a(n+1) = a(n)
- first digit of a(n), ending when 0 is reached.
For example

	S[38] = (38,35,32,29,27,25,23,21,19,18,17,16,15,14,13,12,11,10,9,0).

You can show that for k >= 1, S[10^(k+1)] includes element 10^k, which
implies that S[10^k] is a suffix of S[10^(k+1)].

Now let R[n] be the reverse of S[n], so

	R[38] = (0,9,10,11,12,13,14,15,16,17,18,19,21,23,25,27,29,32,35,38).

R[n] is then a sequence of the type you originally described, with a(n) =
a(n-1) + first digit of a(n), staring at 0 and ending at n.

Also, R[10^k] is a prefix of R[10^(k+1)].

Finally, note that |R[10^k]|  < |R[10^(k+1)]|, so the length of |R[10^k]|
grows without bound as k increases.

Therefore, lim k->inf R[10^k] is an infinite sequence with a(n) = a(n-1) +
first digit of a(n), as required.


> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Eric
> Angelini
> Sent: Wednesday, November 06, 2013 10:29 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Sum of a(n-1) and a(n)'s first digit
> 
> 
> Hello SeqFans,
> Lars has found some fascinating patterns -- see here:
> http://www.cetteadressecomportecinquantesignes.com/FirstDigit.htm
> 
> Many thanks to Lars.
> Best,
> É.
> 
> -----Ursprungligt meddelande-----
> From: Eric Angelini
> Sent: Monday, November 04, 2013 5:22 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Sum of a(n-1) and a(n)'s first digit
> 
> Hello Seqfans,
> let a(n) be the sum of a(n-1) and a(n)'s first digit.
> 
> Example : a(n-1) = 597
>           a(n)   = 603  (as 597 + 6 = 603)
> 
> This doesn't work sometimes:
> 
> Example : a(n-1) = 991
>           a(n) = ?
> 
> Is there an infinite such sequence?
> 
> If you start the sequence with 0,9,... you might be
> stuck: ..., 68, 75, 83, 92, ?
> 
> But there is a way to jump over 100: simply select this
> branch:
> 
> 0,9,10,11,12,13,14,15,16,17,18,20,22,...
> 
> instead of:
> 
> 0,9,10,11,12,13,14,15,16,17,18,19,21,...
> 
> The first branch leads to ..., 68, 75, 83, 92.
> The second one to ...  54, 60, 66, 73, 81, 90, 99, 100, 101, ...
> 
> Best,
> É.
> 
> 
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