A190085 - OEIS

A190085

a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sin(1/2), t=cos(1/2).

1, 3, 6, 8, 11, 13, 16, 18, 20, 22, 25, 27, 30, 32, 35, 37, 39, 41, 44, 46, 49, 51, 54, 56, 57, 60, 62, 65, 67, 70, 72, 75, 76, 79, 81, 84, 86, 89, 91, 94, 95, 98, 100, 103, 105, 108, 110, 113, 115, 116, 119, 121, 124, 126, 129, 131, 134, 135, 138, 140, 143, 145, 148, 150, 153, 154, 157, 159, 162, 164, 167, 169, 171, 173, 175, 178, 180

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let f(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define g(n) and h(n) as the ranks of n/s and n/t. It is easy to prove that

f(n) = n + [n*s/r] + [n*t/r],

g(n) = n + [n*r/s] + [n*t/s],

h(n) = n + [n*r/t] + [n*s/t], where []=floor.

Taking r=1, s=sin(1/2), t=cos(1/2) gives f=A190085, g=A190086, h=A190087.

LINKS

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

FORMULA

A190085: f(n) = n + [n*sin(1/2)] + [n*cos(1/2)], this sequence.

A190086: g(n) = n + [n*csc(1/2)] + [n*cot(1/2)].

A190087: h(n) = n + [n*sec(1/2)] + [n*tan(1/2)].

MATHEMATICA

r=1; s=Sin[1/2]; t=Cos[1/2];