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

Author

N. J. A. Sloane, May 12 2013, based on email from Christopher Carl Heckman, May 06 2013

Keywords

• nonn

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)

Examples

			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
		

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

Crossrefs

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

Programs

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 *)

Extensions

Corrected and edited by Christopher Carl Heckman, May 12 2013

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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