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A189380
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a(n) = n + floor(n*s/r) + floor(n*t/r); r=1, s=-1+sqrt(2), t=1+sqrt(2).
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3
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3, 6, 11, 14, 19, 22, 25, 30, 33, 38, 41, 44, 49, 52, 57, 60, 65, 68, 71, 76, 79, 84, 87, 90, 95, 98, 103, 106, 111, 114, 117, 122, 125, 130, 133, 136, 141, 144, 149, 152, 155, 160, 163, 168, 171, 176, 179, 182, 187, 190, 195, 198, 201, 206, 209, 214, 217, 222, 225, 228, 233, 236, 241, 244, 247, 252, 255, 260, 263, 266, 271, 274, 279, 282, 287, 290, 293, 298, 301, 306, 309, 312, 317
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OFFSET
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1,1
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COMMENTS
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This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
f(n) = n + [n*s/r] + [n*t/r],
g(n) = n + [n*r/s] + [n*t/s],
h(n) = n + [n*r/t] + [n*s/t], where []=floor.
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LINKS
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FORMULA
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a(n) = 5*floor(n * (sqrt(2) - 1)) + 3*floor(n * (2 - sqrt(2))) + 3. - Miko Labalan, Dec 04 2016
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MATHEMATICA
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r=1; s=-1+2^(1/2); t=1+2^(1/2);
f[n_] := n + Floor[n*s/r] + Floor[n*t/r];
g[n_] := n + Floor[n*r/s] + Floor[n*t/s];
h[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[f[n], {n, 1, 120}] (* A189380 *)
Table[g[n], {n, 1, 120}] (* A189381 *)
Table[h[n], {n, 1, 120}] (* A189382 *)
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PROG
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(PARI) for(n=1, 100, print1(n + floor(n*(sqrt(2) -1)) + floor(n*(sqrt(2)+1)), ", ")) \\ G. C. Greubel, Apr 20 2018
(Magma) [n + Floor(n*(Sqrt(2) -1)) + Floor(n*(Sqrt(2) + 1)): n in [1..100]]; // G. C. Greubel, Apr 20 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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