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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 21 2008
STATUS
approved