A225376 - OEIS

%I #34 Mar 05 2023 03:08:25

%S 1,5,11,20,36,60,94,140,199,272,360,465,588,730,893,1078,1286,1519,

%T 1778,2064,2378,2721,3094,3498,3934,4403,4907,5448,6027,6645,7303,

%U 8002,8743,9527,10355,11228

%N 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.

%C P can be extended for 10^6 terms, but it is not known if P,Q,R can be extended to infinity.

%C A probabilistic argument suggests that P, Q, R are infinite. - _N. J. A. Sloane_, May 19 2013

%C 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).

%D M. Gardner, Weird Numbers from Titan, Isaac Asimov's Science Fiction Magazine, Vol. 4, No. 5, May 1980, pp. 42ff.

%H Christopher Carl Heckman, <a href="/A225376/b225376.txt">Table of n, a(n) for n = 1..10002</a>

%e The initial terms of P, Q, R are:

%e 1     5    11    20    36    60    94   140   199   272   360

%e    4     6     9    16    24    34    46    59    73    88

%e       2     3     7     8    10    12    13    14    15

%p Hofstadter2 := proc (N) local h, dh, ddh, S, lbmex, i:

%p     h := 1, 5, 11: dh := 4, 6: ddh := 2:

%p     lbmex := 3: S := {h,dh,ddh}:

%p     for i from 4 to N do:

%p        while lbmex in S do: S := S minus {lbmex}: lbmex := lbmex + 1: od:

%p        ddh := ddh, lbmex:

%p        dh := dh, dh[-1] + lbmex:

%p        h := h, h[-1] + dh[-1]:

%p        S := S union {h[-1], dh[-1], ddh[-1]}:

%p        lbmex := lbmex + 1:

%p     od:

%p     if {h} intersect {dh} <> {} then: return NULL:

%p     elif {h} intersect {ddh} <> {} then: return NULL:

%p     elif {ddh} intersect {dh} <> {} then: return NULL:

%p     else: return [h]: fi:

%p end proc: # _Christopher Carl Heckman_, May 12 2013

%t 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]];

%t Hofstadter2[36] (* _Jean-François Alcover_, Mar 05 2023, after _Christopher Carl Heckman_'s Maple code *)

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_, May 12 2013, based on email from _Christopher Carl Heckman_, May 06 2013

%E Corrected and edited by _Christopher Carl Heckman_, May 12 2013