

A225376


Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,5,11, Q starts with 4,6, R starts with 2; at each stage the smallest number not yet present in P,Q,R is appended to R; every number appears exactly once in the union of P,Q,R. Sequence gives P.


8



1, 5, 11, 20, 36, 60, 94, 140, 199, 272, 360, 465, 588, 730, 893, 1078, 1286, 1519, 1778, 2064, 2378, 2721, 3094, 3498, 3934, 4403, 4907, 5448, 6027, 6645, 7303, 8002, 8743, 9527, 10355, 11228
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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



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:


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]];


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



