OFFSET
1,1
COMMENTS
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
FORMULA
(* 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].
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, May 09 2011
STATUS
approved