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A190372
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a(n) = n + [n*s/r] + [n*t/r] + [n*u/r]; r=sin(Pi/5), s=1/r, t=sin(2*Pi/5), u=1/t.
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4
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5, 13, 20, 28, 35, 42, 50, 57, 65, 71, 78, 86, 94, 101, 108, 115, 123, 131, 136, 144, 151, 159, 167, 173, 181, 189, 196, 204, 209, 217, 225, 232, 240, 247, 254, 262, 269, 275, 283, 290, 298, 305, 312, 320, 327, 335, 343, 348, 356, 363, 371, 379, 385, 393, 400, 408, 414, 421, 429, 437, 444, 451, 458, 466, 474, 481, 487, 495, 502
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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=1/r, t=sin(2*Pi/5), u=1/t gives
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LINKS
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FORMULA
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(* A190372 *) f[n_] := n + Floor[n/sin(Pi/5)^2] + Floor[2*n*cos(Pi/5)] + Floor[n/(sin(2*Pi/5)*sin(Pi/5))].
(* A190373 *) g[n_] := n + Floor[n*sin(Pi/5)^2] + Floor[n*sin(Pi/5)* sin(2*Pi/5)] + Floor[n/(2*cos(Pi/5))].
(* A190374 *) h[n_] := n + Floor[n/(2*cos(Pi/5))] + Floor[n/(sin(Pi/5)* sin(2*Pi/5))] + Floor[n/sin(2*Pi/5)^2].
(* A190375 *) i[n_] := n + Floor[n*sin(Pi/5)*sin(2*Pi/5)] + Floor[2*n*cos(Pi/5)] + Floor[n*sin(2*Pi/5)^2].
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MATHEMATICA
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r=Sin[Pi/5]; s=1/r; t=Sin[2*Pi/5]; u=1/t;
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}] (* A190372 *)
Table[g[n], {n, 1, 120}] (* A190373 *)
Table[h[n], {n, 1, 120}] (* A190374 *)
Table[i[n], {n, 1, 120}] (* A190375 *)
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PROG
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(PARI) for(n=1, 100, print1(n + floor(n/sin(Pi/5)^2) + floor(2*n*cos(Pi/5)) + floor(n/(sin(2*Pi/5)*sin(Pi/5))), ", ")) \\ G. C. Greubel, Apr 05 2018
(Magma) R:=RealField(); [n + Floor(n/Sin(Pi(R)/5)^2) + 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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