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For S a string of numbers, let F(S) = the sum of those numbers. Let a(1)=1. For n>1, a(n) is the largest k such that the string a(1)a(2)...a(n-1) can be written in the form [x][y_1][y_2]...[y_k], where each y_i is positive (but not necessarily all the same) length and F(y_i)=F(y_j) for all i,j<=k.
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%I #16 Feb 08 2023 07:56:57

%S 1,1,2,2,3,1,2,2,3,2,3,2,4,3,3,3,4,1,3,2,2,3,3,3,3,4,2,3,3,4,2,5,2,2,

%T 4,3,2,5,2,3,3,2,4,2,3,3,2,3,3,3,3,4,2,3,3,4,4,4,3,3,4,4,4,3,3,3,3,4,

%U 4,4,4,4,5,3,4,3,2,3,3,2,3,4,4,3,3,5,3,3,3,4,5,3,3,3,4,3,3,5,3,6,3,3,4,6,2

%N For S a string of numbers, let F(S) = the sum of those numbers. Let a(1)=1. For n>1, a(n) is the largest k such that the string a(1)a(2)...a(n-1) can be written in the form [x][y_1][y_2]...[y_k], where each y_i is positive (but not necessarily all the same) length and F(y_i)=F(y_j) for all i,j<=k.

%C Here multiplication denotes concatenation of strings. This is Gijswijt's sequence, A090822, except that the 'y' blocks count as being equivalent whenever the sum of their digits is equal.

%H Rémy Sigrist, <a href="/A092331/b092331.txt">Table of n, a(n) for n = 1..10000</a>

%H F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">A Slow-Growing Sequence Defined by an Unusual Recurrence</a>, J. Integer Sequences, Vol. 10 (2007), #07.1.2.

%H F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [<a href="http://neilsloane.com/doc/gijs.pdf">pdf</a>, <a href="http://neilsloane.com/doc/gijs.ps">ps</a>].

%H Rémy Sigrist, <a href="/A092331/a092331.txt">C program</a>

%H <a href="/index/Ge#Gijswijt">Index entries for sequences related to Gijswijt's sequence</a>

%e From _Rémy Sigrist_, Feb 08 2023: (Start)

%e The first terms, alongside an appropriate partition of prior terms, are:

%e n a(n) Prior terms

%e -- ---- -----------------

%e 1 1 N/A

%e 2 1 1

%e 3 2 1|1

%e 4 2 1 1|2

%e 5 3 1 1|2|2

%e 6 1 1 1 2 2 3

%e 7 2 1 1|2 2|3 1

%e 8 2 1 1 2 2|3 1 2

%e 9 3 1 1|2 2|3 1|2 2

%e 10 2 1|1 2 2 3|1 2 2 3

%e (End)

%o (C) See Links section.

%Y Cf. A090822, A091975, A091976.

%K nonn

%O 1,3

%A J. Taylor (integersfan(AT)yahoo.com), Mar 17 2004