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A225385 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives P.

%I A225385 #23 Mar 06 2023 09:12:13
%S A225385 1,3,9,20,38,64,100,148,209,284,374,480,603,745,908,1093,1301,1533,
%T A225385 1790,2074,2386,2727,3098,3500,3934,4401,4902,5438,6011,6623,7275,
%U A225385 7968,8703,9481,10303,11170,12083,13043,14052,15111,16221,17383,18598,19867,21191,22571,24008,25503,27057,28671,30347,32086,33890,35760,37697,39702,41776,43920
%N A225385 Construct sequences P,Q,R by the rules: Q = first differences of P, R = second differences of P, P starts with 1,3,9, Q starts with 2,6, R starts with 4; at each stage the smallest number not yet present in P,Q,R is appended to R. Sequence gives P.
%C A225385 In contrast to A225376-A225378, here it is not required (and not true) that each number should appear just once in P union Q union R. On the other hand, again in contrast to A225376-A225378, here it is obvious that P, Q, R are infinite.
%C A225385 The first three numbers that are repeated are 284, 2074, 3500, which appear in both P and Q. There may be no others. Of course R is disjoint from P and Q, by definition.
%p A225385 # Based on _Christopher Carl Heckman_'s program for A225376.
%p A225385 f:=proc(N) local h,dh,ddh,S,mex,i;
%p A225385 h:=1,3,9; dh:=2,6; ddh:=4; mex:=5; S:={h,dh,ddh};
%p A225385 for i from 4 to N do
%p A225385 while mex in S do S:=S minus {mex}; mex:=mex+1; od;
%p A225385 ddh:=ddh,mex; dh:=dh,dh[-1]+mex; h:=h,h[-1]+dh[-1];
%p A225385 S:=S union {h[-1], dh[-1], ddh[-1]};
%p A225385 mex:=mex+1;
%p A225385 od;
%p A225385 RETURN([[h],[dh],[ddh]]);
%p A225385 end;
%p A225385 f(100);
%t A225385 f[N_] := Module[{P = {1, 3, 9}, Q = {2, 6}, R = {4}, S, mex = 5, i},
%t A225385   S = Join[P, Q, R];
%t A225385   For[i = 4, i <= N, i++,
%t A225385    While[MemberQ[S, mex], S = S~Complement~{mex}; mex++];
%t A225385    AppendTo[R, mex];
%t A225385    AppendTo[Q, Q[[-1]] + mex];
%t A225385    AppendTo[P, P[[-1]] + Q[[-1]]];
%t A225385    S = S~Union~{P[[-1]], Q[[-1]], R[[-1]]}; mex++];
%t A225385 P];
%t A225385 f[100] (* _Jean-François Alcover_, Mar 06 2023, after Maple code *)
%Y A225385 Cf. A225386, A225387, A005228, A030124, A037257, A225376, A225377, A225378.
%K A225385 nonn
%O A225385 1,2
%A A225385 _N. J. A. Sloane_, May 15 2013