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%I #22 Dec 19 2021 04:17:37
%S 1,2,1,3,1,1,2,4,1,1,1,2,1,2,3,5,1,1,1,1,2,1,1,2,1,2,3,1,2,3,4,6,1,1,
%T 1,1,1,2,1,1,1,2,1,1,2,1,2,3,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,7,1,1,1,1,
%U 1,1,2,1,1,1,1,2,1,1,1,2,1,1,2,1,2,3,1,1,1,2,1,1,2,1,2,3,1,1,2,1,2,3,1,2,3,4,1,1,2,1,2,3,1,2,3,4,1,2,3,4
%N a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.
%C If all 1's are deleted, the remaining terms are the sequence incremented. - after _Franklin T. Adams-Watters_ Oct 05 2006 comment in A051135.
%C Ordinal transform of A162598.
%H Antti Karttunen, <a href="/A265332/b265332.txt">Table of n, a(n) for n = 1..8192</a>
%H 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.
%F a(1) = 1; for n > 1, a(n) = A051135(n).
%e Illustration how the sequence can be constructed by concatenating the frequency counts Q_n of each successive level n of A004001-tree:
%e --
%e 1 Q_0 = (1)
%e |
%e _2__ Q_1 = (2)
%e / \
%e _3 __4_____ Q_2 = (1,3)
%e / / | \
%e _5 _6 _7 __8___________ Q_3 = (1,1,2,4)
%e / / / | / | \ \
%e _9 10 11 12 13 14 15___ 16_________ Q_4 = (1,1,1,2,1,2,3,5)
%e / / / / | / / | |\ \ | \ \ \ \
%e 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
%e --
%e The above illustration copied from the page 229 of Kubo and Vakil paper (page 5 in PDF).
%t terms = 120;
%t h[1] = 1; h[2] = 1;
%t h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
%t seq[nmax_] := seq[nmax] = (Length /@ Split[Sort @ Array[h, nmax, 2]])[[;; terms]];
%t seq[nmax = 2 terms];
%t seq[nmax += terms];
%t While[seq[nmax] != seq[nmax - terms], nmax += terms];
%t seq[nmax] (* _Jean-François Alcover_, Dec 19 2021 *)
%o (Scheme) (define (A265332 n) (if (= 1 n) 1 (A051135 n)))
%Y Essentially same as A051135 apart from the initial term, which here is set as a(1)=1.
%Y Cf. A004001, A265901, A265903.
%Y Cf. A162598 (corresponding other index).
%Y Cf. A265754.
%Y Cf. also A267108, A267109, A267110.
%K nonn
%O 1,2
%A _Antti Karttunen_, Jan 09 2016
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