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A298875
Solution (b(n)) of the system of 3 equations in Comments.
4
2, 6, 11, 18, 26, 35, 45, 57, 70, 84, 99, 116, 135, 155, 176, 198, 221, 245, 270, 298, 327, 357, 388, 420, 453, 487, 523, 560, 598, 637, 677, 718, 760, 804, 850, 897, 945, 994, 1044, 1095, 1147, 1200, 1254, 1309, 1365, 1423, 1482, 1543, 1605, 1668, 1732
OFFSET
0,1
COMMENTS
Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new;
b(n) = a(n) + b(n-1);
c(n) = a(n) + 2 b(n);
where "least new k" means the least positive integer not yet placed.
***
Do these sequences a,b,c partition the positive integers? They differ from the corresponding partitioning sequences A298871, A298872, and A298872. For example, A298872(56) = 2139, whereas A298875(56) = 2138.
LINKS
EXAMPLE
n: 0 1 2 3 4 5 6 7 8 9
a: 1 4 5 7 8 9 10 12 13 14
b: 2 6 11 18 26 35 45 57 70 84
c: 3 16 27 43 60 30 79 100 126 153
MATHEMATICA
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
AppendTo[b, Last[a] + Last[b]],
AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
Take[a, 100] (*A298874*)
Take[b, 100] (*A298875*)
Take[c, 100] (*A298876*)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 19 2018
STATUS
approved