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A190508 a(n) = n+[n*s/r]+[n*t/r]+[n*u/r]; r=golden ratio, s=r^2, t=r^3, u=r^4.

%I A190508 #12 Feb 15 2025 03:06:57
%S A190508 8,18,26,36,47,55,65,73,84,94,102,112,123,131,141,149,160,170,178,188,
%T A190508 196,207,217,225,235,246,254,264,272,283,293,301,311,322,330,340,348,
%U A190508 358,369,377,387,395,406,416,424,434,445,453,463,471,482,492,500,510,518,529,539,547,557,568,576,586,594,605,615,623,633,644
%N A190508 a(n) = n+[n*s/r]+[n*t/r]+[n*u/r]; r=golden ratio, s=r^2, t=r^3, u=r^4.
%C A190508 This is one of four sequences that partition the positive integers. In general, suppose that r, s, t, u are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1}, {h/u: h>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the four sets are jointly ranked. Define b(n), c(n), d(n) as the ranks of n/s, n/t, n/u, respectively.
%C A190508 It is easy to prove that
%C A190508   a(n)=n+[n*s/r]+[n*t/r]+[n*u/r],
%C A190508   b(n)=n+[n*r/s]+[n*t/s]+[n*u/s],
%C A190508   c(n)=n+[n*r/t]+[n*s/t]+[n*u/t],
%C A190508   d(n)=n+[n*r/u]+[n*s/u]+[n*t/u], where []=floor.
%C A190508 Taking r=golden ratio, s=r^2, t=r^3, u=r^4 gives a=A190508, b=A190509, c=A054770, d=A190511.
%F A190508 A190508: a(n)=n+[n*r]+[n*r^2]+[n*r^3]
%F A190508 A190509: b(n)=[n/r]+n+[n*r]+[n*r^2]
%F A190508 A054770: c(n)=[n/r^2]+[n/r]+n+[n*r]
%F A190508 A190511: d(n)=[n/r^3]+[n/r^2]+[n/r]+n
%t A190508 r=GoldenRatio; s=r^2; t=r^3; u=r^4;
%t A190508 a[n_] := n + Floor[n*s/r] + Floor[n*t/r]+Floor[n*u/r];
%t A190508 b[n_] := n + Floor[n*r/s] + Floor[n*t/s]+Floor[n*u/s];
%t A190508 c[n_] := n + Floor[n*r/t] + Floor[n*s/t]+Floor[n*u/t];
%t A190508 d[n_] := n + Floor[n*r/u] + Floor[n*s/u]+Floor[n*t/u];
%t A190508 Table[a[n], {n, 1, 120}]  (*A190508*)
%t A190508 Table[b[n], {n, 1, 120}]  (*A190509*)
%t A190508 Table[c[n], {n, 1, 120}]  (*A054770*)
%t A190508 Table[d[n], {n, 1, 120}]  (*A190511*)
%Y A190508 Cf. A190509, A054770, A190511 (the other three sequences in the partition of N).
%K A190508 nonn
%O A190508 1,1
%A A190508 _Clark Kimberling_, May 11 2011