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A190368 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). 4

%I #26 Sep 08 2022 08:45:57

%S 3,8,12,17,21,26,30,35,39,44,48,53,57,62,66,71,75,80,84,89,93,98,103,

%T 107,112,116,121,125,129,134,139,143,148,152,157,161,165,170,175,180,

%U 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

%N 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).

%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=cos(Pi/5), t=sin(2*Pi/5), u=cos(2*Pi/5) gives

%C f=A190368, g=A190369, h=A190370, i=A190371.

%H G. C. Greubel, <a href="/A190368/b190368.txt">Table of n, a(n) for n = 1..10000</a>

%F A190368: f(n) = n + floor(n*cot(Pi/5)) + floor(2*n*cos(Pi/5)) + floor(n*cos(2*Pi/5)/sin(Pi/5)).

%F A190369: g(n) = n + floor(n*tan(Pi/5)) + floor(2*n*sin(Pi/5)) + floor(n*cos(2*Pi/5)/cos(Pi/5)).

%F A190370: h(n) = n + floor(n*sec(Pi/5)/2) + floor(n*csc(Pi/5)/2) + floor(n*cot(2*Pi/5)).

%F 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)).

%p 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

%t r=Sin[Pi/5]; s=Cos[Pi/5]; t=Sin[2*Pi/5]; u=Cos[2*Pi/5];

%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}] (* A190368 *)

%t Table[g[n], {n, 1, 120}] (* A190369 *)

%t Table[h[n], {n, 1, 120}] (* A190370 *)

%t Table[i[n], {n, 1, 120}] (* A190371 *)

%o (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

%o (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

%Y Cf. A190369, A190370, A190371.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 09 2011

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