OFFSET
1,2
COMMENTS
P can be extended for 10^6 terms, but it is not known if P,Q,R can be extended to infinity.
A probabilistic argument suggests that P, Q, R are infinite. - N. J. A. Sloane, May 19 2013
Martin Gardner (see reference) states that no such triple P,Q,R of sequences exists if it is required that P(1)<Q(1)<R(1).
REFERENCES
M. Gardner, Weird Numbers from Titan, Isaac Asimov's Science Fiction Magazine, Vol. 4, No. 5, May 1980, pp. 42ff.
LINKS
Christopher Carl Heckman, Table of n, a(n) for n = 1..10002
EXAMPLE
The initial terms of P, Q, R are:
1 5 11 20 36 60 94 140 199 272 360
4 6 9 16 24 34 46 59 73 88
2 3 7 8 10 12 13 14 15
MAPLE
Hofstadter2 := proc (N) local h, dh, ddh, S, lbmex, i:
h := 1, 5, 11: dh := 4, 6: ddh := 2:
lbmex := 3: S := {h, dh, ddh}:
for i from 4 to N do:
while lbmex in S do: S := S minus {lbmex}: lbmex := lbmex + 1: od:
ddh := ddh, lbmex:
dh := dh, dh[-1] + lbmex:
h := h, h[-1] + dh[-1]:
S := S union {h[-1], dh[-1], ddh[-1]}:
lbmex := lbmex + 1:
od:
if {h} intersect {dh} <> {} then: return NULL:
elif {h} intersect {ddh} <> {} then: return NULL:
elif {ddh} intersect {dh} <> {} then: return NULL:
else: return [h]: fi:
end proc: # Christopher Carl Heckman, May 12 2013
MATHEMATICA
Hofstadter2[N_] := Module[{P, Q, R, S, k, i}, P = {1, 5, 11}; Q = {4, 6}; R = {2}; k = 3; S = Join[P, Q, R]; For[i = 4, i <= N, i++, While[MemberQ[S, k], S = S~Complement~{k}; k++]; AppendTo[R, k]; AppendTo[Q, Q[[-1]] + k]; AppendTo[P, P[[-1]] + Q[[-1]]]; S = S~Union~{P[[-1]], Q[[-1]], R[[-1]]}; k++]; Which[P~Intersection~Q != {}, Return@Nothing, {P}~Intersection~R != {}, Return@Nothing, R~Intersection~Q != {}, Return@Nothing, True, Return@P]];
Hofstadter2[36] (* Jean-François Alcover, Mar 05 2023, after Christopher Carl Heckman's Maple code *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, May 12 2013, based on email from Christopher Carl Heckman, May 06 2013
EXTENSIONS
Corrected and edited by Christopher Carl Heckman, May 12 2013
STATUS
approved