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.

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

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

Cf. A225377, A225378, A005228, A030124, A037257.

Sequence in context: A026038 A080957 A118375 * A099400 A139534 A245773

Adjacent sequences: A225373 A225374 A225375 * A225377 A225378 A225379

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