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A299641
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Solution (d(n)) of the system of 5 complementary equations in Comments.
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5
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4, 8, 13, 17, 21, 25, 30, 34, 38, 42, 47, 51, 55, 59, 64, 68, 72, 76, 81, 85, 89, 93, 98, 102, 106, 110, 115, 119, 123, 127, 132, 136, 140, 144, 149, 153, 157, 161, 166, 170, 174, 178, 183, 187, 191, 195, 200, 204, 208, 212, 217, 221, 225, 229, 234, 238, 242
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OFFSET
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0,1
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COMMENTS
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Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:
a(n) = least new;
b(n) = least new;
c(n) = least new;
d(n) = least new;
e(n) = a(n) + b(n) + c(n) + d(n);
where "least new k" means the least positive integer not yet placed.
***
Conjecture: for all n >= 0,
0 <= 17n - 11 - 4 a(n) <= 4
0 <= 17n - 7 - 4 b(n) <= 4
0 <= 17n - 3 - 4 c(n) <= 3
0 <= 17n + 1 - 4 d(n) <= 3
0 <= 17n - 5 - e(n) <= 3
***
The sequences a,b,c,d,e partition the positive integers. The sequence e can be called the "anti-tetranacci sequence"; see A075326 (anti-Fibonacci numbers) and A265389 (anti-tribonacci numbers).
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LINKS
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EXAMPLE
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n: 0 1 2 3 4 5 6 7 8 9
a: 1 5 9 14 18 22 27 31 35 39
b: 2 6 11 15 19 23 28 32 36 40
c: 3 7 12 16 20 24 29 33 37 41
d: 4 8 13 17 21 25 30 34 38 42
e: 10 26 45 62 78 94 114 130 146 162
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MATHEMATICA
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z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
Last[a] + Last[b] + Last[c] + Last[d]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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