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%I #8 Jan 27 2016 12:23:21
%S 1,2,3,4,7,5,6,8,9,17,10,11,12,13,14,15,16,31,18,19,20,21,22,23,24,25,
%T 49,74,26,27,28,29,30,32,33,34,35,36,71,37,38,39,40,41,42,43,44,45,46,
%U 47,48,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,127,65,66,67,68,69,70,72,73,75,76,77,78,79,80,81,161
%N If a(n) is a square (and n > 1), then a(n+1) = a(n) + a(n-1), else a(n+1) is the smallest positive integer not occurring earlier.
%C A variant of the sequence A267758 where the relation has to hold for prime numbers rather than for squares.
%C Conjectured to be a permutation of the positive integers (which could be enforced by definition). In case there would occur a duplicate, it must be of the form a(n+1) = a(n) + a(n-1) and equal to an earlier term a(m+1) of the same form, where furthermore the predecessor a(m-1) also is of that form, since otherwise a(m+1) would be smaller than this a(n+1). This seems extremely unlikely to happen, and maybe provably impossible.
%H M. F. Hasler, <a href="/A268132/b268132.txt">Table of n, a(n) for n = 1..1000</a>
%e a(26) = 25 is a square, thus followed by a(26) + a(25) = 25 + 24 = 49 which is again a square, thus followed by 49 + 25 = 74. Where is the next occurrence of two subsequent squares?
%o (PARI) a(n,show=0,is=x->issquare(x),a=[1],L=0,U=[])={while(#a<n,show&&if(type(show)=="t_STR",write(show,#a," ",a[#a]),print1(a[#a]",")); if(a[#a]>L+1,U=setunion(U,[a[#a]]),L++;while(#U&&U[1]<=L+1,U=U[^1];L++));a=concat(a,if(!is(a[#a])||#a<2,L+1,a[#a]+a[#a-1])));if(type(show)=="t_VEC",a,a[#a])}
%K nonn
%O 1,2
%A _M. F. Hasler_, Jan 26 2016
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