

A304497


Solution (a(n)) of the system of complementary equations defined in Comments.


6



1, 3, 6, 8, 10, 13, 15, 17, 20, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 116, 118, 120, 122, 125, 127, 129, 132, 134, 136, 139, 141
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OFFSET

0,2


COMMENTS

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1:
a(n) = least new,
b(n) = least new,
c(n) = 2*a(n) + b(n),
where "least new k" means the least positive integer not yet placed. The three sequences partition the positive integers. Empirically, for all n >= 0,
1 <= 3*a(n)  7*n <= 5,
5 <= 3*b(n)  7*n <= 8,
3 <= c(n)  7*n <= 6.


LINKS



EXAMPLE

a(0) = 1, b(0) = 2; c(0) = 2*1 + 2 = 4, so that a(1) = 3, so that b(1) = 4, so that c(1) = 11.


MATHEMATICA

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
AppendTo[c, 2 Last[a] + Last[b]], {z}];
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment > ".", Dividers > {{2 > Red, 1 > Blue}, {2 > Red, 1 > Blue}}]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



