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A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments.

%I A298468 #20 Jan 22 2022 23:41:44
%S A298468 1,4,6,9,12,14,16,19,21,24,26,29,32,34,36,39,42,44,46,49,52,54,56,59,
%T A298468 61,64,66,69,72,74,76,79,81,84,86,89,92,94,96,99,101,104,106,109,112,
%U A298468 114,116,119,122,124,126,129,132,134,136,139,141,144,146,149
%N A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments.
%C A298468 Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:
%C A298468 aa(n) = least new;
%C A298468 bb(n) = aa(n) + cc(n-1);
%C A298468 cc(n) = least new;
%C A298468 where "least new k" means the least positive integer not yet placed.
%C A298468 ***
%C A298468 The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >=0,
%C A298468 (1) 5*n - 1 - 2*aa(n) is in {0,1,2},
%C A298468 (2) (aa(n) mod 5) is in {1,2,4},
%C A298468 (3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;
%C A298468 (4) (bb(n) mod 5) is in {1,2}.
%C A298468 From _N. J. A. Sloane_, Nov 05 2019: (Start)
%C A298468 Conjecture: For t >= 1, aa(2t) = 5t+1(+1 if binary expansion of t ends in an odd number of 0's), and for t >= 0, aa(2t+1) = 5t+4.
%C A298468 The first part may also be written as aa(2t) = 5t+1+A328789(t-1).
%C A298468 (End)
%H A298468 Clark Kimberling, <a href="/A298468/b298468.txt">Table of n, a(n) for n = 0..10000</a> [This is aa]
%e A298468 n:  0 1 2 3 4 5 6 7 8 9 10
%e A298468 aa: 1 4 6 9 12 14 16 19 21 24 26
%e A298468 bb: 2 7 11 17 22 27 31 37 41 47 51
%e A298468 cc: 3 5 8 10 13 15 18 20 23 25 28
%t A298468 z = 500;
%t A298468 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A298468 a = {1}; b = {2}; c = {3};
%t A298468 Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
%t A298468   AppendTo[b, Last[a] + Last[c]];
%t A298468   AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
%t A298468 Take[a, 100] (* A298468 *)
%t A298468 Take[b, 100] (* A297469 *)
%t A298468 Take[c, 100] (* A047218 *)
%t A298468 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A298468 Cf. A299634, A297469 (bb), A047218 (cc), A328789.
%K A298468 nonn,easy
%O A298468 0,2
%A A298468 _Clark Kimberling_, May 04 2018
%E A298468 Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - _N. J. A. Sloane_, Nov 03 2019