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A190053 a(n) = n + [n*s/r] + [n*t/r]; r=2, s=sin(pi/3), t=csc(pi/3). 4
1, 3, 5, 7, 9, 11, 14, 15, 17, 19, 21, 23, 25, 28, 29, 31, 33, 35, 37, 39, 42, 43, 45, 47, 49, 52, 53, 56, 57, 59, 61, 63, 66, 67, 70, 71, 74, 75, 77, 80, 81, 84, 85, 88, 89, 91, 94, 95, 98, 99, 102, 104, 105, 108, 109, 112, 113, 116, 118, 119, 122, 123, 126, 127, 130, 132, 134, 136, 137, 140, 141, 144, 146, 148, 150, 151, 154, 156, 158, 160 (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 a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that

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

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

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

Taking r=2, s=sin(pi/3), t=csc(pi/3) gives

a=A190053, b=A190054, c=A190055.

LINKS

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

FORMULA

A190053:  a(n) = n + [(n/2)*sin(pi/3)] + [(n/2)*csc(pi/3)].

A190054:  b(n) = n + [2n*csc(pi/3)] + [n*(csc(pi/3))^2].

A190055:  c(n) = n + [2n*sin(pi/3)] + [n*(sin(pi/3))^2].

MATHEMATICA

r=2; s=Sin[Pi/3]; t=Csc[Pi/3];

a[n_] := n + Floor[n*s/r] + Floor[n*t/r];

b[n_] := n + Floor[n*r/s] + Floor[n*t/s];

c[n_] := n + Floor[n*r/t] + Floor[n*s/t];

Table[a[n], {n, 1, 120}]  (*A190053*)

Table[b[n], {n, 1, 120}]  (*A190054*)

Table[c[n], {n, 1, 120}]  (*A190055*)

PROG

(PARI) for(n=1, 100, print1(n + floor(n*sin(Pi/3)/2) + floor(n/(2*sin(Pi/3))), ", ")) \\ G. C. Greubel, Jan 10 2018

(MAGMA) C<i> := ComplexField(); [n + Floor(n*Sin(Pi(C)/3)/2) + Floor(n/(2*Sin(Pi(C)/3))): n in [1..100]]; // G. C. Greubel, Jan 10 2018

CROSSREFS

Cf. A190054, A190055.

Sequence in context: A024323 A118820 A117521 * A285590 A195179 A033038

Adjacent sequences:  A190050 A190051 A190052 * A190054 A190055 A190056

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 04 2011

STATUS

approved

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Last modified August 17 17:06 EDT 2019. Contains 326059 sequences. (Running on oeis4.)