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A189932 a(n) = n + [n*r/s] + [n*t/s]; r=1, s=(sin(Pi/5))^2, t=(cos(Pi/5))^2, where [] denotes the floor function. 4
4, 10, 16, 22, 28, 34, 40, 46, 52, 56, 62, 68, 74, 80, 86, 92, 98, 104, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 166, 172, 178, 184, 190, 196, 202, 208, 214, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 276, 282, 288, 294, 300, 306, 312, 318, 324, 328, 334, 340, 346, 352, 358, 364, 370, 376, 382, 386, 392, 398, 404 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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=1, s=(sin(Pi/5))^2, t=cos((Pi/5))^2 gives a=A005408, b=A189932, c=A189933.

LINKS

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

FORMULA

From G. C. Greubel, Jan 13 2018: (Start)

A005408:  a(n) = n + [n*(sin(Pi/5))^2] + [n*(cos(Pi/5))^2] = 2*n - 1.

A189932:  b(n) = n + [n*(csc(Pi/5))^2] + [n*(cot(Pi/5))^2].

A189933:  c(n) = n + [n*(sec(Pi/5))^2] + [n*(tan(Pi/5))^2]. (End)

MATHEMATICA

r=1; s=Sin[Pi/5]^2; t=Cos[Pi/5]^2;

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}]  (*A005408*)

Table[b[n], {n, 1, 120}]  (*A189932*)

Table[c[n], {n, 1, 120}]  (*A189933*)

Table[b[n]/2, {n, 1, 120}]  (*A189934*)

Table[c[n]/2, {n, 1, 120}]  (*A189935*)

PROG

(PARI) for(n=1, 100, print1(n + floor(n/(sin(Pi/5))^2) + floor(n/(tan(Pi/5))^2), ", ")) \\ G. C. Greubel, Jan 13 2018

(MAGMA) C<i> := ComplexField(); [n + Floor(n/(Sin(Pi(C)/5))^2) + Floor(n/(Tan(Pi(C)/5))^2): n in [1..100]]; // G. C. Greubel, Jan 13 2018

CROSSREFS

Cf. A189933, A189934, A189926.

Sequence in context: A310530 A310531 A310532 * A310533 A269960 A016957

Adjacent sequences:  A189929 A189930 A189931 * A189933 A189934 A189935

KEYWORD

nonn

AUTHOR

Clark Kimberling, May 01 2011

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)