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%I #8 Sep 08 2022 08:45:57
%S 5,13,20,28,35,42,50,57,65,71,78,86,94,101,108,115,123,131,136,144,
%T 151,159,167,173,181,189,196,204,209,217,225,232,240,247,254,262,269,
%U 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
%N 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.
%C 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
%C f(n) = n + [n*s/r] + [n*t/r] + [n*u/r],
%C g(n) = n + [n*r/s] + [n*t/s] + [n*u/s],
%C h(n) = n + [n*r/t] + [n*s/t] + [n*u/t],
%C i(n) = n + [n*r/u] + [n*s/u] + [n*t/u], where []=floor.
%C Taking r=sin(Pi/5), s=1/r, t=sin(2*Pi/5), u=1/t gives
%C f=A190372, g=A190373, h=A190374, i=A190375.
%H G. C. Greubel, <a href="/A190372/b190372.txt">Table of n, a(n) for n = 1..10000</a>
%F (* 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))].
%F (* 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))].
%F (* 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].
%F (* 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].
%t r=Sin[Pi/5]; s=1/r; t=Sin[2*Pi/5]; u=1/t;
%t f[n_] := n + Floor[n*s/r] + Floor[n*t/r] + Floor[n*u/r];
%t g[n_] := n + Floor[n*r/s] + Floor[n*t/s] + Floor[n*u/s];
%t h[n_] := n + Floor[n*r/t] + Floor[n*s/t] + Floor[n*u/t];
%t i[n_] := n + Floor[n*r/u] + Floor[n*s/u] + Floor[n*t/u];
%t Table[f[n], {n, 1, 120}] (* A190372 *)
%t Table[g[n], {n, 1, 120}] (* A190373 *)
%t Table[h[n], {n, 1, 120}] (* A190374 *)
%t Table[i[n], {n, 1, 120}] (* A190375 *)
%o (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
%o (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
%Y Cf. A190373, A190374, A190375.
%K nonn
%O 1,1
%A _Clark Kimberling_, May 09 2011
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