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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
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, 35, 37, 38, 39, 40, 42, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 65, 67, 68, 69, 71, 72, 74, 75, 76, 78, 79, 80, 81, 82, 84, 85, 87, 88, 89, 90, 91, 93, 94 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D Hanna, Table of n, a(n) for n = 1..420

FORMULA

CONJECTURES on evaluating limits.

The following limits exist for some irrational q and r:

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

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

Thus q and r can be defined by:

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

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

Therefore q = least positive real root that satisfies:

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

Also, r = least positive real root that satisfies:

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

EXAMPLE

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

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

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

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

Union of Y and X+Y = positive integers:

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

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

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

PROG

(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))))))}

CROSSREFS

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

Sequence in context: A286924 A039100 A184482 * A160992 A096997 A187480

Adjacent sequences:  A141201 A141202 A141203 * A141205 A141206 A141207

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 21 2008

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.