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%I #10 May 01 2018 03:00:54
%S 2,8,10,18,24,26,33,35,42,45,54,56,63,66,74,76,82,88,94,96,102,105,
%T 114,116,123,125,134,136,142,145,154,156,162,168,170,178,180,186,194,
%U 196,202,208,214,216,222,225,234,236,243,246,254,256,262,265,274,276
%N Solution (c(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="/A297293/b297293.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, A297291, A297292.
%K nonn,easy
%O 0,1
%A _Clark Kimberling_, Apr 24 2018
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