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A162598 Ordinal transform of A265332.

%I A162598 #22 Dec 19 2021 08:31:47
%S A162598 1,1,2,1,3,4,2,1,5,6,7,3,8,4,2,1,9,10,11,12,5,13,14,6,15,7,3,16,8,4,2,
%T A162598 1,17,18,19,20,21,9,22,23,24,10,25,26,11,27,12,5,28,29,13,30,14,6,31,
%U A162598 15,7,3,32,16,8,4,2,1,33,34,35,36,37,38,17,39,40,41,42,18,43,44,45,19,46,47
%N A162598 Ordinal transform of A265332.
%C A162598 This is a fractal sequence.
%C A162598 It appears that each group of 2^k terms starts with 1 and ends with the remaining powers of two from 2^k down to 2^1.
%C A162598 From _Antti Karttunen_, Jan 09-12 2016: (Start)
%C A162598 This is ordinal transform of A265332, which is modified A051135 (with a(1) = 1, instead of 2). -  after _Franklin T. Adams-Watters_' original definition for this sequence.
%C A162598 A000079 (powers of 2) indeed gives the positions of ones in this sequence. This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 of PDF), which together also imply the pattern observed above, more clearly represented as:
%C A162598 a(2) = 1.
%C A162598 a(3..4) = 2, 1.
%C A162598 a(6..8) = 4, 2, 1.
%C A162598 a(13..16) = 8, 4, 2, 1.
%C A162598 a(28..31) = 16, 8, 4, 2, 1.
%C A162598 etc.
%C A162598 (End)
%H A162598 Antti Karttunen, <a href="/A162598/b162598.txt">Table of n, a(n) for n = 1..8192</a>
%H A162598 T. Kubo and R. Vakil, <a href="http://dx.doi.org/10.1016/0012-365X(94)00303-Z">On Conway's recursive sequence</a>, Discr. Math. 152 (1996), 225-252.
%H A162598 <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>
%F A162598 Let b(1) = 1, b(n) = A051135(n) for n > 1. Then a(n) is the number of b(k) that equal b(n) for 1 <= k <= n: sum( 1, 1<=k<=n and a(k)=a(n) ).
%F A162598 If A265332(n) = 1, then a(n) = A004001(n), otherwise a(n) = a(A080677(n)-1) = a(n - A004001(n)). - _Antti Karttunen_, Jan 09 2016
%t A162598 terms = 100;
%t A162598 h[1] = 1; h[2] = 1;
%t A162598 h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
%t A162598 t = Array[h, 2*terms];
%t A162598 A051135 = Take[Transpose[Tally[t]][[2]], terms];
%t A162598 b[_] = 1;
%t A162598 a[n_] := a[n] = With[{t = If[n == 1, 1, A051135[[n]]]}, b[t]++];
%t A162598 Array[a, terms] (* _Jean-François Alcover_, Dec 19 2021, after _Robert G. Wilson v_ in A051135 *)
%o A162598 (Scheme, with memoization-macro definec)
%o A162598 (definec (A162598 n) (if (= 1 (A265332 n)) (A004001 n) (A162598 (- (A080677 n) 1))))
%o A162598 ;; _Antti Karttunen_, Jan 09 2016
%Y A162598 Cf. A004001, A051135, A080677, A087686.
%Y A162598 Row index of A265901, column index of A265903.
%Y A162598 Cf. A265332 (corresponding other index).
%Y A162598 Cf. A000079 (positions of ones).
%Y A162598 Cf. A000225 (from the term 3 onward the positions of 2's).
%Y A162598 Cf. A000325 (from its third term 5 onward the positions of 3's, which occur always as the last term before the next descending subsequence of powers of two).
%K A162598 nonn,look
%O A162598 1,3
%A A162598 _Franklin T. Adams-Watters_, Jul 07 2009
%E A162598 Name amended by _Antti Karttunen_, Jan 09 2016