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A189527
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a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sqrt(3)-sqrt(2), t=sqrt(3)+sqrt(2).
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3
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4, 8, 12, 17, 21, 25, 31, 35, 39, 44, 48, 52, 57, 62, 66, 71, 75, 79, 84, 88, 93, 97, 102, 106, 110, 115, 119, 124, 129, 133, 137, 142, 146, 150, 156, 160, 164, 169, 173, 177, 182, 187, 191, 195, 200, 204, 208, 214, 218, 222, 227, 231, 235, 240, 245, 249, 254, 258, 262, 267, 271, 276, 281, 285, 289, 293, 298, 302, 307, 312, 316, 320, 325, 329, 333, 339, 343, 347, 352, 356, 360, 365, 370
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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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MATHEMATICA
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r=1; s=3^(1/2)-2^(1/2); t=3^(1/2)+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}] (* A189527 *)
Table[g[n], {n, 1, 120}] (* A189528 *)
Table[h[n], {n, 1, 120}] (* A189529 *)
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PROG
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(PARI) for(n=1, 100, print1(n + floor(n*(sqrt(3) - sqrt(2))) + floor(n*(sqrt(3)+sqrt(2))), ", ")) \\ G. C. Greubel, Apr 20 2018
(Magma) [n + Floor(n*(Sqrt(3) - Sqrt(2))) + Floor(n*(Sqrt(3)+Sqrt(2))): 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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