A190372 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.

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

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 f=A190372, g=A190373, h=A190374, i=A190375.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

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

Cf. A190373, A190374, A190375.

Sequence in context: A055045 A213741 A184825 * A184837 A197120 A319449

Adjacent sequences: A190369 A190370 A190371 * A190373 A190374 A190375

Keyword

nonn

Author

Clark Kimberling, May 09 2011

Status

approved