[seqfan] Re: Sum of a(n-1) and a(n)'s first digit
David Wilson
davidwwilson at comcast.net
Thu Nov 14 04:13:26 CET 2013
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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