OFFSET
1,2
COMMENTS
Since the empty set and all the singletons except {0} have the required property, a(n) >= n. And clearly a(n) <= 2^n, since there are only 2^n possible subsets. - Michael B. Porter, Feb 11 2012
EXAMPLE
a(5) = 7 because the following 7 subsets of {0,1,2,3,4} have the required property:
1: { }
2: { 1 }
3: { 1, 4 }
4: { 2 }
5: { 2, 3 }
6: { 3 }
7: { 4 }
MAPLE
b:= proc(i, n, s) local si; si:= s union {i};
`if`(i=0, 1, b(i-1, n, s) +`if`({seq(seq(irem(k+j, n)
, j=si), k=si)} intersect si={}, b(i-1, n, si), 0))
end:
a:= n-> b(n-1, n, {}):
seq(a(n), n=1..25); # Alois P. Heinz, Apr 24 2012
MATHEMATICA
b[i_, n_, s_] := Module[{si = s ~Union~ {i}}, If[i == 0, 1, b[i-1, n, s] + If[ Flatten[ Table[ Table[ Mod[k+j, n], {j, si}], {k, si}]] ~ Intersection~ si == {}, b[i-1, n, si], 0]]]; a[n_] := a[n] = b[n-1, n, {}]; Table[ Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 40}] (* Jean-François Alcover, Jun 07 2013, translated and adapted from Alois P. Heinz's Maple program *)
PROG
(Haskell)
import Control.Monad
--this creates the powerset of a set
ps n = filterM (\x->[True, False]) n
--given a set z, this creates the set X of (a+b) for all a, b, in Z
addset z = do x<-z
y<-z
[x+y]
--this check if two sets are disjoint
disjoint a [] = True
disjoint a (c:d) = (disjoint a d) && ((filter (\x->x==c) a) ==[])
--this checks if a set z is disjoint from its "adsset" in a certain Zn, n being the second argument.
good z n = disjoint z (map (\x->rem x n) (addset z))
--this generates all off Zn's subsets with the required property.
sets n = filter (\x ->good x n) (ps [0..(n-1)])
--this generates the first n terms of the sequence
sequence n = map (\x->length(sets x) ) [1..n]
(PARI) a(n)=if(n<4, return(n)); my(u, v=vector(n-2, i, [i]), s=n, t); while(#v, u=List(); for(i=1, #v, t=v[i]; for(m=t[#t]+1, n, if(setsearch(t, 2*m%n)==0 && #setintersect(Set(vector(#t, k, t[k]+m)%n), t)==0 && #setintersect(vector(#t, k, m-t[#t-k+1]), t)==0, listput(u, concat(t, m))))); v=Vec(u); s+=#v); s \\ Charles R Greathouse IV, Jul 31 2016
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Dan Fodor, Feb 11 2012
EXTENSIONS
More terms from Joerg Arndt, Feb 11 2012
a(31)-a(43) from Alois P. Heinz, Apr 24 2012
STATUS
approved