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.

1, 3, 9, 20, 38, 64, 100, 148, 209, 284, 374, 480, 603, 745, 908, 1093, 1301, 1533, 1790, 2074, 2386, 2727, 3098, 3500, 3934, 4401, 4902, 5438, 6011, 6623, 7275, 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

Offset: 1

Author

N. J. A. Sloane, May 15 2013

Keywords

• nonn

Comments

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.

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.

Crossrefs

Cf. A225386, A225387, A005228, A030124, A037257, A225376, A225377, A225378.

Programs

Maple

# Based on Christopher Carl Heckman's program for A225376.
f:=proc(N) local h,dh,ddh,S,mex,i;
h:=1,3,9; dh:=2,6; ddh:=4; mex:=5; S:={h,dh,ddh};
for i from 4 to N do
while mex in S do S:=S minus {mex}; mex:=mex+1; od;
ddh:=ddh,mex; dh:=dh,dh[-1]+mex; h:=h,h[-1]+dh[-1];
S:=S union {h[-1], dh[-1], ddh[-1]};
mex:=mex+1;
od;
RETURN([[h],[dh],[ddh]]);
end;
f(100);

Mathematica

f[N_] := Module[{P = {1, 3, 9}, Q = {2, 6}, R = {4}, S, mex = 5, i},
  S = Join[P, Q, R];
  For[i = 4, i <= N, i++,
   While[MemberQ[S, mex], S = S~Complement~{mex}; mex++];
   AppendTo[R, mex];
   AppendTo[Q, Q[[-1]] + mex];
   AppendTo[P, P[[-1]] + Q[[-1]]];
   S = S~Union~{P[[-1]], Q[[-1]], R[[-1]]}; mex++];
P];
f[100] (* Jean-François Alcover, Mar 06 2023, after Maple code *)