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%I #15 May 14 2021 04:40:10
%S 1,2,3,4,9,14,19,24,53,82,111,140,309,478,647,816,1801,2786,3771,4756,
%T 10497,16238,21979,27720,61181,94642,128103,161564,356589,551614,
%U 746639,941664,2078353,3215042,4351731,5488420,12113529,18738638,25363747
%N Start with the sequence S(0)={1,1} and for k>0 define S(k) to be I(S(k-1)) where I denotes the operation of inserting, for i=1,2,3..., the term a(i)+a(i+1) between any two terms for which 4a(i+1)<=5a(i). The listed terms are the initial terms of the limit of this process as k goes to infinity.
%C Conjectures:
%C (1) the section (a(2n+1)}={1,3,9,19,53,111,...} is A077442, the terms of which are solutions of ax^2+7 = a square,
%C (2) the section {a(4n+1)}={1,9,53,309,1801,...} is A038761,
%C (3) the section {a(4n+2)}={2,14,82,478,2786,...} is A077444, the terms of which are solutions of 2x^2+8 = a square,
%C (4) the sequence {a(4n+2)/2}={1,7,41,239,1393,...} is A002315, the terms of which are solutions of 2x^2+2 = a square,
%C (5) the section {a(4n+4)}={4,24,140,816,4756,...} is A005319, the terms of which are solutions of 2x^2+4=a square,
%C (6) the sequence {a(4n+4)/4}={1,6,35,204,1189,...} is A001109, the terms of which are solutions of 8x^2+1=a square.
%H Ivan Neretin, <a href="/A082981/b082981.txt">Table of n, a(n) for n = 1..1000</a>
%H John W. Layman, <a href="http://www.math.vt.edu/people/layman/sequences/ins_seq.htm">Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types</a>, June 2003 [Broken link]
%H John W. Layman, <a href="/A085376/a085376.txt">Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types</a>, June 2003 [local copy, corrected]
%H John W. Layman, <a href="https://intranet.math.vt.edu/people/layman/sequences/agedetit.htm">Sequences Generated by Age-Determined Insertion Trees</a>, Jan 2006
%H John W. Layman, <a href="/A117535/a117535.txt">Sequences Generated by Age-Determined Insertion Trees</a>, Jan 2006 [Local copy]
%F It appears that a(n)=6a(n-4)-a(n-8).
%F Empirical g.f.: x*(x+1)^2*(x^2+1)^2/((x^4-2*x^2-1)*(x^4+2*x^2-1)). - _Colin Barker_, Nov 06 2014
%t Most@Nest[If[#[[-2]] >= 4 #[[-1]], Append[Most@#, #[[-1]] + #[[-2]]], Insert[#, #[[-1]] + #[[-2]], -2]] &, {1, 1}, 47] (* _Ivan Neretin_, Apr 27 2017 *)
%Y Cf. A001109, A002315, A005319, A038761, A077442, A077444.
%K nonn
%O 1,2
%A _John W. Layman_, May 28 2003
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