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A265332 a(n) is the index of the column in A265901 where n appears; also the index of the row in A265903 where n appears.

%I A265332 #22 Dec 19 2021 04:17:37
%S A265332 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 A265332 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 A265332 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 A265332 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 A265332 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 A265332 Ordinal transform of A162598.
%H A265332 Antti Karttunen, <a href="/A265332/b265332.txt">Table of n, a(n) for n = 1..8192</a>
%H A265332 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 A265332 a(1) = 1; for n > 1, a(n) = A051135(n).
%e A265332 Illustration how the sequence can be constructed by concatenating the frequency counts Q_n of each successive level n of A004001-tree:
%e A265332 --
%e A265332              1                                      Q_0 = (1)
%e A265332              |
%e A265332             _2__                                    Q_1 = (2)
%e A265332            /    \
%e A265332          _3    __4_____                             Q_2 = (1,3)
%e A265332         /     /  |     \
%e A265332       _5    _6  _7    __8___________                Q_3 = (1,1,2,4)
%e A265332      /     /   / |   /  |  \        \
%e A265332    _9    10  11 12  13  14  15___    16_________    Q_4 = (1,1,1,2,1,2,3,5)
%e A265332   /     /   /  / |  /  / |   |\  \   | \  \  \  \
%e A265332 17    18  19 20 21 22 23 24 25 26 27 28 29 30 31 32
%e A265332 --
%e A265332 The above illustration copied from the page 229 of Kubo and Vakil paper (page 5 in PDF).
%t A265332 terms = 120;
%t A265332 h[1] = 1; h[2] = 1;
%t A265332 h[n_] := h[n] = h[h[n - 1]] + h[n - h[n - 1]];
%t A265332 seq[nmax_] := seq[nmax] = (Length /@ Split[Sort @ Array[h, nmax, 2]])[[;; terms]];
%t A265332 seq[nmax = 2 terms];
%t A265332 seq[nmax += terms];
%t A265332 While[seq[nmax] != seq[nmax - terms], nmax += terms];
%t A265332 seq[nmax] (* _Jean-François Alcover_, Dec 19 2021 *)
%o A265332 (Scheme) (define (A265332 n) (if (= 1 n) 1 (A051135 n)))
%Y A265332 Essentially same as A051135 apart from the initial term, which here is set as a(1)=1.
%Y A265332 Cf. A004001, A265901, A265903.
%Y A265332 Cf. A162598 (corresponding other index).
%Y A265332 Cf. A265754.
%Y A265332 Cf. also A267108, A267109, A267110.
%K A265332 nonn
%O A265332 1,2
%A A265332 _Antti Karttunen_, Jan 09 2016