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A296484
Solution (a(n)) of the system of 3 complementary equations in Comments.
3
1, 3, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76
OFFSET
0,2
COMMENTS
Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
a(n) = least new;
b(n) = a(n-1)+c(n-1);
c(n) = 2 a(n) + b(n);
where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.
LINKS
EXAMPLE
n: 0 1 2 3 4 5 6 7 8 9
a: 1 3 6 7 8 9 10 12 13 15
b: 2 5 14 32 53 77 104 134 170 209
c: 4 11 26 46 69 95 124 158 196 239
MATHEMATICA
z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
AppendTo[b, a[[n - 1]] + c[[n - 1]]],
AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A296484 *)
Take[b, 100] (* A296502 *)
Take[c, 100] (* A297149 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 24 2018
STATUS
approved