A154328 - OEIS

%I #25 Mar 08 2024 01:12:40

%S 1,10,11,12,20,111,112,120,1000,1001,1002,1003,1004,1005,1006,1007,

%T 1008,1009,1010,1011,1012,1013,1014,1015,1016,10000,

%U 10000000000000000000,10000000800000000000,10000000800000000001,10000000800000000002

%N Lexicographically earliest strictly increasing sequence having the property that a(n) is the sum of the first a(n) digits of the sequence.

%C The variant where the condition of strict monotonicity is dropped is less straightforward to compute.

%C The sequence could also be encoded in a more compact way by specifying only the indices n where it jumps (a(n) > a(n-1)+1) and the corresponding values a(n), see A154329-A154330.

%H M. F. Hasler, <a href="/A154328/b154328.txt">Table of n, a(n) for n = 1..73</a>

%H Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/SelfSum.htm">An ugly self-describing sequence</a>.

%H E. Angelini, <a href="/A154328/a154328.pdf">An ugly self-describing sequence</a> [Cached copy, with permission]

%H E. Angelini et al., <a href="http://list.seqfan.eu/oldermail/seqfan/2009-January/000478.html">"Ugly digit sum", SeqFan mailing list, Jan 08 2009</a>

%e Starting with a(1)=1, the next term a(2) > a(1) cannot be 2,...,9 (else the sum of these digits would be larger): the least possibility not leading to a contradiction is a(2)=10.

%e Then we can go on with a(3)=11, a(4)=12, but a(5) cannot be 13, the least possibility is a(5)=20.

%e See the linked web page for more details and sequences A154329-A154330 for terms beyond those given here.

%o (PARI) /* Note: This code checks only whether there is a contradiction for the given digits (1st arg), it does not ensure minimality. If the 2nd arg is nonzero, it dumps a list of all digits and partial sums. */

%o check_A154328(S=[1,10,11,12,20],dump=0)={

%o local(d=eval(Vec(concat(concat([""],S)))),t=0,ds=vector(#d,i,t+=d[i]));

%o dump && print(vector(#d,i,Str(i":"d[i]":"ds[i])));

%o for(i=1,#S, S[i]>#d && break; ds[S[i]]==S[i] || error("wrong at i=",i,": [S[i],ds[S[i]]]=",[S[i],ds[S[i]]]));

%o print("no contradiction for terms <= "#d) }

%Y Cf. A155817.

%K nonn,base

%O 1,2

%A _Eric Angelini_ and _M. F. Hasler_, Jan 13 2009