%I A141204
%S A141204 1,2,4,5,6,8,9,10,11,13,14,15,17,18,20,21,22,23,24,26,27,29,30,31,33,34,
%T A141204 35,37,38,39,40,42,43,45,46,47,49,50,51,52,53,55,56,58,59,60,61,62,64,
%U A141204 65,67,68,69,71,72,74,75,76,78,79,80,81,82,84,85,87,88,89,90,91,93,94
%N A141204 Let sequences X and Y consist of the least positive integers such that 
               2X+Y is the complement of X and X+Y is the complement of Y, starting 
               with X(1)=1 and Y(1)=1; then this sequence equals X, while 2X+Y=A141205, 
               Y=A141206 and X+Y=A141207.
%H A141204 Paul D Hanna, <a href="b141204.txt">Table of n, a(n) for n = 1..420</
               a>
%F A141204 CONJECTURES on evaluating limits.
%F A141204 The following limits exist for some irrational q and r:
%F A141204 Limit X(n)/n = 1 + q, Limit {2X+Y}(n)/n = 1 + 1/q and
%F A141204 Limit Y(n)/n = 1 + r, Limit {X+Y}(n)/n = 1 + 1/r.
%F A141204 Thus q and r can be defined by:
%F A141204 Limit X(n)/{2X+Y}(n) = q = (1 + q)/(3 + 2*q + r) and
%F A141204 Limit Y(n)/{X+Y}(n) = r = (1 + r)/(2 + r + q).
%F A141204 Therefore q = least positive real root that satisfies:
%F A141204 1 - 4*q + 2*q^2 + 2*q^3 = 0, giving q = 0.31544880690757230308868993...
%F A141204 Also, r = least positive real root that satisfies:
%F A141204 2 - 4*r + r^3 = 0, giving r = 0.5391888728108891165258759...
%e A141204 Union of X and 2X+Y = positive integers:
%e A141204 X=[1,2,4,5,6,8,9,10,11,13,14,15,17,18,20,21,22,23,24,...];
%e A141204 2X+Y=[3,7,12,16,19,25,28,32,36,41,44,48,54,57,63,66,70,...].
%e A141204 Limit X(n)/{2X+Y}(n) = 0.3154488069...
%e A141204 Union of Y and X+Y = positive integers:
%e A141204 Y=[1,3,4,6,7,9,10,12,14,15,16,18,20,21,23,24,26,27,29,...];
%e A141204 X+Y=[2,5,8,11,13,17,19,22,25,28,30,33,37,39,43,45,48,50,...].
%e A141204 Limit Y(n)/{X+Y}(n) = 0.5391888728...
%o A141204 (PARI) /* Print a(n), n=1..100: */ {A=[1]; B=[3]; C=[1]; D=[2]; print1(A[1]",
               "); for(n=1, 100, for(j=2, 4*n, if(setsearch(Set(concat(A, B)), j)==0, 
               At=concat(A, j); for(k=2*j+1, 6*n, if(setsearch(Set(concat(At, B)), 
               k)==0, if(setsearch(Set(concat(C, D)), k-2*j)==0, if(setsearch(Set(concat(C, 
               D)), k-j)==0, A=At; B=concat(B, k); C=concat(C, k-2*j); D=concat(D, 
               k-j); print1(A[ #A]","); break); break))))))}
%Y A141204 Cf. A141205 (2X+Y), A141206 (Y), A141207 (X+Y).
%Y A141204 Sequence in context: A030124 A064318 A039100 this_sequence A160992 A096997 
               A081694
%Y A141204 Adjacent sequences: A141201 A141202 A141203 this_sequence A141205 A141206 
               A141207
%K A141204 nonn
%O A141204 1,2
%A A141204 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 21 2008