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%I #4 Apr 25 2018 08:32:51
%S 1,4,5,9,12,13,16,17,21,22,27,28,31,32,37,38,41,44,47,48,51,52,57,58,
%T 61,62,67,68,71,72,77,78,81,84,85,89,90,93,97,98,101,104,107,108,111,
%U 112,117,118,121,122,127,128,131,132,137,138,141,144,147,148
%N Solution (a(n)) of the system of 3 complementary equations in Comments.
%C Define sequences a(n), b(n), c(n) recursively:
%C a(n) = least new;
%C b(n) = least new > = a(n) + 2;
%C c(n) = a(n) + b(n) - 2;
%C where "least new k" means the least positive integer not yet placed.
%C ***
%C The sequences a,b,c partition the positive integers.
%C ***
%C Conjectures: for n >=0,
%C 0 <= 5*n + 4 - 2*a(n) <= 5,
%C 0 <= 5*n + 8 - 2*b(n) <= 4,
%C 0 <= c(n) - 5n <= 4.
%H Clark Kimberling, <a href="/A297291/b297291.txt">Table of n, a(n) for n = 0..1000</a>
%e n: 0 1 2 3 4 5 6 7 8 9 10
%e a: 1 4 5 9 12 13 16 17 21 27 28
%e b: 3 6 7 11 14 15 19 20 23 25 29
%e c: 2 8 10 18 24 26 33 35 42 45 54
%t z = 300;
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a = b = c = {};
%t Do[{AppendTo[a,
%t mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
%t AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
%t Take[a, 100] (* A297291 *)
%t Take[b, 100] (* A297292 *)
%t Take[c, 100] (* A297293 *)
%t (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y Cf. A299634, A297292, A297293.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Apr 24 2018
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