%I #18 Sep 08 2022 08:45:56

%S 4,10,16,22,28,34,40,46,52,56,62,68,74,80,86,92,98,104,108,114,120,

%T 126,132,138,144,150,156,162,166,172,178,184,190,196,202,208,214,218,

%U 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

%N 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.

%C 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

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

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

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

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

%H G. C. Greubel, <a href="/A189932/b189932.txt">Table of n, a(n) for n = 1..10000</a>

%F From _G. C. Greubel_, Jan 13 2018: (Start)

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

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

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

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

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

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

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

%t Table[a[n], {n, 1, 120}] (*A005408*)

%t Table[b[n], {n, 1, 120}] (*A189932*)

%t Table[c[n], {n, 1, 120}] (*A189933*)

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

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

%o (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

%o (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

%Y Cf. A189933, A189934, A189926.

%K nonn

%O 1,1

%A _Clark Kimberling_, May 01 2011