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A190006 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sin(Pi/3), t=csc(Pi/3).

%I A190006 #16 Oct 20 2024 01:42:04
%S A190006 2,5,8,11,14,17,21,23,26,29,32,35,39,42,44,47,50,53,56,60,63,66,68,71,
%T A190006 74,78,81,84,87,89,92,95,99,102,105,108,111,113,117,120,123,126,129,
%U A190006 132,134,138,141,144,147,150,153,157,159,162,165,168,171,174,178,180,183,186,189,192,196,199,202,204,207,210,213,217,220
%N A190006 a(n) = n + [n*s/r] + [n*t/r]; r=1, s=sin(Pi/3), t=csc(Pi/3).
%C A190006 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 A190006   a(n) = n + [n*s/r] + [n*t/r],
%C A190006   b(n) = n + [n*r/s] + [n*t/s],
%C A190006   c(n) = n + [n*r/t] + [n*s/t], where []=floor.
%C A190006 Taking r=1, s=sin(Pi/3), t=csc(Pi/3) gives a=A190006, b=A190007, c=A190008.
%H A190006 G. C. Greubel, <a href="/A190006/b190006.txt">Table of n, a(n) for n = 1..10000</a>
%F A190006 A190006:  a(n) = n + [n*sin(Pi/3)] + [n*csc(Pi/3)].
%F A190006 A190007:  b(n) = n + [n*csc(Pi/3)] + [n*(csc(Pi/3))^2].
%F A190006 A190008:  c(n) = n + [n*sin(Pi/3)] + [n*(sin(Pi/3))^2].
%t A190006 r=1; s=Sin[Pi/3]; t=Csc[Pi/3];
%t A190006 a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
%t A190006 b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
%t A190006 c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
%t A190006 Table[a[n], {n, 1, 120}]  (* A190006 *)
%t A190006 Table[b[n], {n, 1, 120}]  (* A190007 *)
%t A190006 Table[c[n], {n, 1, 120}]  (* A190008 *)
%o A190006 (PARI) for(n=1,100, print1(n + floor(n*sin(Pi/3)) + floor(n/sin(Pi/3)), ", ")) \\ _G. C. Greubel_, Jan 11 2018
%o A190006 (Magma) C<i> := ComplexField(); [n + Floor(n*Sin(Pi(C)/3)) + Floor(n/Sin(Pi(C)/3)): n in [1..100]]; // _G. C. Greubel_, Jan 11 2018
%Y A190006 Cf. A180007, A190008.
%K A190006 nonn,easy
%O A190006 1,1
%A A190006 _Clark Kimberling_, May 03 2011