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

1,2

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.

Links

Table of n, a(n) for n=1..58.

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 *)

Crossrefs

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

Sequence in context: A037048 A348090 A139142 * A037257 A145068 A293357

Adjacent sequences: A225382 A225383 A225384 * A225386 A225387 A225388

Keyword

nonn

Author

N. J. A. Sloane, May 15 2013

Status

approved