A189460 - OEIS

%I #8 Nov 16 2013 13:31:21

%S 3,7,10,14,18,21,25,28,33,37,40,44,48,51,55,58,63,67,70,74,77,81,85,

%T 88,92,97,100,104,107,111,115,118,122,127,130,134,137,141,145,148,152,

%U 155,160,164,167,171,175,178,182,185,190,194,197,201,204,208,212,215,219,224,227,231,234,238,242,245,249,254,257,261,264,268,272,275,279,282,287,291,294,298,302,305,309

%N n+[ns/r]+[nt/r]; r=2, s=sqrt(5), t=1+sqrt(5).

%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=2, s=sqrt(5), t=1+sqrt(5) gives

%C a=A189460, b=A189461, c=A189462.

%H Ivan Panchenko, <a href="/A189460/b189460.txt">Table of n, a(n) for n = 1..10000</a>

%t r=2; s=5^(1/2); t=1+5^(1/2);

%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}]  (*A189460*)

%t Table[b[n], {n, 1, 120}]  (*A189461*)

%t Table[c[n], {n, 1, 120}]  (*A189462*)

%Y Cf. A189461, A189462.

%K nonn

%O 1,1

%A _Clark Kimberling_, Apr 22 2011