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 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. 4

%I

%S 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,

%T 34,35,37,38,39,40,42,43,45,46,47,49,50,51,52,53,55,56,58,59,60,61,62,

%U 64,65,67,68,69,71,72,74,75,76,78,79,80,81,82,84,85,87,88,89,90,91,93,94

%N 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 Paul D Hanna, <a href="/A141204/b141204.txt">Table of n, a(n) for n = 1..420</a>

%F CONJECTURES on evaluating limits.

%F The following limits exist for some irrational q and r:

%F Limit X(n)/n = 1 + q, Limit {2X+Y}(n)/n = 1 + 1/q and

%F Limit Y(n)/n = 1 + r, Limit {X+Y}(n)/n = 1 + 1/r.

%F Thus q and r can be defined by:

%F Limit X(n)/{2X+Y}(n) = q = (1 + q)/(3 + 2*q + r) and

%F Limit Y(n)/{X+Y}(n) = r = (1 + r)/(2 + r + q).

%F Therefore q = least positive real root that satisfies:

%F 1 - 4*q + 2*q^2 + 2*q^3 = 0, giving q = 0.31544880690757230308868993...

%F Also, r = least positive real root that satisfies:

%F 2 - 4*r + r^3 = 0, giving r = 0.5391888728108891165258759...

%e Union of X and 2X+Y = positive integers:

%e X=[1,2,4,5,6,8,9,10,11,13,14,15,17,18,20,21,22,23,24,...];

%e 2X+Y=[3,7,12,16,19,25,28,32,36,41,44,48,54,57,63,66,70,...].

%e Limit X(n)/{2X+Y}(n) = 0.3154488069...

%e Union of Y and X+Y = positive integers:

%e Y=[1,3,4,6,7,9,10,12,14,15,16,18,20,21,23,24,26,27,29,...];

%e X+Y=[2,5,8,11,13,17,19,22,25,28,30,33,37,39,43,45,48,50,...].

%e Limit Y(n)/{X+Y}(n) = 0.5391888728...

%o (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 Cf. A141205 (2X+Y), A141206 (Y), A141207 (X+Y).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 21 2008

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Last modified October 14 17:39 EDT 2019. Contains 328022 sequences. (Running on oeis4.)