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

2, 5, 8, 11, 13, 17, 19, 22, 25, 28, 30, 33, 37, 39, 43, 45, 48, 50, 53, 57, 59, 63, 65, 67, 71, 74, 76, 79, 82, 85, 87, 91, 94, 97, 100, 102, 105, 108, 111, 113, 115, 119, 122, 126, 128, 130, 133, 135, 139, 142, 145, 148, 150, 154, 156, 160, 163, 165, 168, 171, 173

Offset

1,1

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

Crossrefs

Cf. A141204 (X), A141205 (2X+Y), A141206 (Y).

Sequence in context: A292643 A376954 A140101 * A190057 A190305 A294391

Adjacent sequences: A141204 A141205 A141206 * A141208 A141209 A141210

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2008

Status

approved