A141206 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 Y, while X=A141204, 2X+Y=A141205 and X+Y=A141207.

1, 3, 4, 6, 7, 9, 10, 12, 14, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 49, 51, 52, 54, 55, 56, 58, 60, 61, 62, 64, 66, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 83, 84, 86, 88, 89, 90, 92, 93, 95, 96, 98, 99, 101, 103, 104, 106, 107

Offset

1,2

Comments

Complement of A141207.

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(C[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(C[ #C]", "); break); break))))))}

Crossrefs

Cf. A141204 (X), A141205 (2X+Y), A141207 (X+Y).

Sequence in context: A103877 A072561 A330176 * A140100 A292642 A249117

Adjacent sequences: A141203 A141204 A141205 * A141207 A141208 A141209

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2008

Status

approved