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A079668 Start with 1; at n-th step, write down what is in the sequence so far.

%I A079668 #35 Aug 25 2025 16:53:29
%S A079668 1,1,1,3,1,4,1,0,2,1,3,1,0,6,1,1,2,2,3,1,4,2,0,10,1,3,2,3,3,2,4,0,5,1,
%T A079668 6,4,0,12,1,6,2,6,3,3,4,1,5,2,6,0,7,0,8,0,9,1,10,8,0,15,1,8,2,8,3,5,4,
%U A079668 2,5,5,6,1,7,1,8,1,9,2,10,0,11,1,12,10
%N A079668 Start with 1; at n-th step, write down what is in the sequence so far.
%C A079668 After 1 1 1 3 1, we see "4 1's, 0 2's and 1 3", so next terms are 4 1 0 2 3 1.
%H A079668 Alois P. Heinz, <a href="/A079668/b079668.txt">Table of n, a(n) for n = 1..20377</a> (first 55 steps)
%e A079668 Row n lists all terms written at the n-th step:
%e A079668   1;
%e A079668   1, 1;
%e A079668   3, 1;
%e A079668   4, 1,  0, 2, 1, 3;
%e A079668   1, 0,  6, 1, 1, 2, 2, 3, 1, 4;
%e A079668   2, 0, 10, 1, 3, 2, 3, 3, 2, 4, 0, 5, 1, 6;
%e A079668   4, 0, 12, 1, 6, 2, 6, 3, 3, 4, 1, 5, 2, 6, 0, 7, 0, 8, 0, 9, 1, 10;
%e A079668   ...
%p A079668 b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+add(x^i, i=T(n))) end:
%p A079668 T:= proc(n) option remember; `if`(n=1, 1, (p->
%p A079668       seq([coeff(p, x, i), i][], i=ldegree(p)..degree(p)))(b(n-1)))
%p A079668     end:
%p A079668 seq(T(n), n=1..10);  # _Alois P. Heinz_, Aug 24 2025
%t A079668 s={1}; Do[s=Flatten[{s,{Count[s,#],#}&/@Range[Min[s],Max[s]]}],{20}];s (* _Peter J. C. Moses_, Mar 21 2013 *)
%o A079668 (Python)
%o A079668 from itertools import islice
%o A079668 def agen(): # generator of terms
%o A079668     a = [1]
%o A079668     yield 1
%o A079668     while True:
%o A079668         counts = []
%o A079668         for d in range(min(a), max(a)+1):
%o A079668             c = a.count(d)
%o A079668             counts.extend([c, d])
%o A079668         a += counts
%o A079668         yield from counts
%o A079668 print(list(islice(agen(), 84))) # _Michael S. Branicky_, Aug 24 2025
%Y A079668 For other versions see A051120, A055186, A079686.
%Y A079668 Cf. A030717, A333867.
%K A079668 nonn,base,tabf,changed
%O A079668 1,4
%A A079668 _N. J. A. Sloane_ Jan 26 2003
%E A079668 More terms from _Sean A. Irvine_, Aug 24 2025