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%I #22 Jan 17 2016 13:56:38
%S 1,2,2,3,3,3,4,5,5,5,5,6,6,7,8,9,9,9,9,9,10,10,10,11,11,12,13,13,14,
%T 15,16,17,17,17,17,17,17,18,18,18,18,19,19,19,20,20,21,22,22,22,23,23,
%U 24,25,25,26,27,28,28,29,30,31,32,33,33,33,33,33,33,33,34,34,34,34,34,35,35,35
%N a(n) = n + 1 - A004001(n).
%C From _Antti Karttunen_, Jan 10 2016: (Start)
%C This is the sequence b(n) mentioned on page 229 (page 5 of PDF) in Kubo & Vakil paper, but using starting offset 1 instead of 2.
%C The recursive sum formula for A004001, a(n) = a(a(n-1)) + a(n-a(n-1)) can be written also as a(n) = a(a(n-1)) + a(A080677(n-1)).
%C This is the least monotonic left inverse for sequence A087686. Proof: Taking the first differences of this sequence yields the characteristic function for the complement of A188163, because A188163 gives the positions where A004001 increases, and this sequence increases by one whenever A004001 does not increase (and vice versa). Sequence A188163 is also 1 followed by A088359 (see comment in former), whose complement A087686 is, thus A087686 is also the complement of A188163, apart from the initial one. Note also how A087686 is closed with respect to A004001 (see A266188).
%C (End)
%D J. Arkin, D. C. Arney, L. S. Dewald and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
%H Antti Karttunen, <a href="/A080677/b080677.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.
%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>
%F a(n) = n + 1 - A004001(n).
%F Other identities. For all n >= 1:
%F a(A087686(n)) = n. [See comments.] - _Antti Karttunen_, Jan 10 2016
%o (Scheme) (define (A080677 n) (- (+ 1 n) (A004001 n))) ;; Scheme code for A004001 given in that entry. - _Antti Karttunen_, Jan 10 2016
%Y Cf. A004001, A087686, A088359, A162598, A188163, A265332, A266188, A267111.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Mar 03 2003