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A190368
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a(n) = n + [n*s/r] + [n*t/r] + [n*u/r]; r=sin(Pi/5), s=cos(Pi/5), t=sin(2*Pi/5), u=cos(2*Pi/5).
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4
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3, 8, 12, 17, 21, 26, 30, 35, 39, 44, 48, 53, 57, 62, 66, 71, 75, 80, 84, 89, 93, 98, 103, 107, 112, 116, 121, 125, 129, 134, 139, 143, 148, 152, 157, 161, 165, 170, 175, 180, 184, 188, 193, 198, 201, 207, 211, 216, 220, 224, 229, 234, 237, 243, 246, 252, 256, 260, 266, 270, 274, 279, 283, 288, 293, 296, 302, 306, 310, 315, 319, 324
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OFFSET
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1,1
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COMMENTS
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This is one of four sequences that partition the positive integers. In general, suppose that r, s, t, u are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1, {h/u: h>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the four sets are jointly ranked. Define b(n), c(n), d(n) as the ranks of n/s, n/t, n/u, respectively. It is easy to prove that
f(n) = n + [n*s/r] + [n*t/r] + [n*u/r],
g(n) = n + [n*r/s] + [n*t/s] + [n*u/s],
h(n) = n + [n*r/t] + [n*s/t] + [n*u/t],
i(n) = n + [n*r/u] + [n*s/u] + [n*t/u], where []=floor.
Taking r=sin(Pi/5), s=cos(Pi/5), t=sin(2*Pi/5), u=cos(2*Pi/5) gives
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LINKS
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FORMULA
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A190368: f(n) = n + floor(n*cot(Pi/5)) + floor(2*n*cos(Pi/5)) + floor(n*cos(2*Pi/5)/sin(Pi/5)).
A190369: g(n) = n + floor(n*tan(Pi/5)) + floor(2*n*sin(Pi/5)) + floor(n*cos(2*Pi/5)/cos(Pi/5)).
A190370: h(n) = n + floor(n*sec(Pi/5)/2) + floor(n*csc(Pi/5)/2) + floor(n*cot(2*Pi/5)).
A190371: i(n) = n + floor(n*sin(Pi/5)/cos(2*Pi/5)) + floor(n*cos(Pi/5)/cos(2*Pi/5)) + floor(n*tan(2*Pi/5)).
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MAPLE
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r:=sin(Pi/5): s:=cos(Pi/5): t:=sin(2*Pi/5): u:=cos(2*Pi/5): seq(n+floor(n*s/r)+floor(n*t/r)+floor(n*u/r), n=1..80); # Muniru A Asiru, Apr 08 2018
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MATHEMATICA
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r=Sin[Pi/5]; s=Cos[Pi/5]; t=Sin[2*Pi/5]; u=Cos[2*Pi/5];
f[n_] := n + Floor[n*s/r] + Floor[n*t/r] + Floor[n*u/r];
g[n_] := n + Floor[n*r/s] + Floor[n*t/s] + Floor[n*u/s];
h[n_] := n + Floor[n*r/t] + Floor[n*s/t] + Floor[n*u/t];
i[n_] := n + Floor[n*r/u] + Floor[n*s/u] + Floor[n*t/u];
Table[f[n], {n, 1, 120}] (* A190368 *)
Table[g[n], {n, 1, 120}] (* A190369 *)
Table[h[n], {n, 1, 120}] (* A190370 *)
Table[i[n], {n, 1, 120}] (* A190371 *)
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PROG
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(PARI) for(n=1, 100, print1(n + floor(n/tan(Pi/5)) + floor(2*n*cos(Pi/5)) + floor(n*cos(2*Pi/5)/sin(Pi/5)), ", ")) \\ G. C. Greubel, Apr 05 2018
(Magma) R:=RealField(); [n + Floor(n/Tan(Pi(R)/5)) + Floor(2*n*Cos(Pi(R)/5)) + Floor(n*Cos(2*Pi(R)/5)/Sin(Pi(R)/5)): n in [1..100]]; // G. C. Greubel, Apr 05 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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