|
%I #12 Sep 17 2015 10:19:25
%S 6,14,22,30,38,46,54,62,70,78,86,94,102,110,118,126,134,142,150,158,
%T 166,174,182,190,198,206,214,222,230,238,246,254,262,270,278,286,294,
%U 302,310,318,326,334,342,350,358,366,374,380,388,396,404,412,420,428,436,444,452,460,468,476,484,492,500,508,516,524,532,540,548
%N a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).
%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+[ns/r]+[nt/r],
%C b(n)=n+[nr/s]+[nt/s],
%C c(n)=n+[nr/t]+[ns/t], where []=floor.
%C Taking r=Pi/2, s=arcsin(5/13), t=arcsin(12/13) gives
%C a=A005408, b=A189785, c=A189786. Note that r=s+t.
%C a(n) first differs from A017137(n-1) at n=48 (a(48)=380 but A017137(47)=382). - _Nathaniel Johnston_, May 16 2011
%t r=Pi/2; s=ArcSin[5/13]; t=ArcSin[12/13];
%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}] (*A189785*)
%t Table[c[n], {n, 1, 120}] (*A189786*)
%t Table[b[n]/2, {n, 1, 120}] (*A189787*)
%t Table[c[n]/2, {n, 1, 120}] (*A004773*)
%Y Cf. A189786, A189787, A004773.
%K nonn
%O 1,1
%A _Clark Kimberling_, Apr 27 2011
|
