A190508 n+[ns/r]+[nt/r]+[nu/r]; r=golden ratio, s=r^2, t=r^3, u=r^4.

8, 18, 26, 36, 47, 55, 65, 73, 84, 94, 102, 112, 123, 131, 141, 149, 160, 170, 178, 188, 196, 207, 217, 225, 235, 246, 254, 264, 272, 283, 293, 301, 311, 322, 330, 340, 348, 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

Offset

1,1

Comments

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. It is easy to prove that

a(n)=n+[ns/r]+[nt/r]+[nu/r],

b(n)=n+[nr/s]+[nt/s]+[nu/s],

c(n)=n+[nr/t]+[ns/t]+[nu/t],

d(n)=n+[nr/u]+[ns/u]+[nt/u], where []=floor.

Taking r=golden ratio, s=r^2, t=r^3, u=r^4 gives

a=A190508, b=A190509, c=A054770, d=A190511.

Links

Table of n, a(n) for n=1..68.

Formula

A190508: a(n)=n+[nr]+[nr^2]+[nr^3]

A190509: b(n)=[n/r]+n+[nr]+[nr^2]

A054770: c(n)=[n/r^2]+[n/r]+n+[nr]

A190511: d(n)=[n/r^3]+[n/r^2]+[n/r]+n

Mathematica

r=GoldenRatio; s=r^2; t=r^3; u=r^4;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r]+Floor[n*u/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s]+Floor[n*u/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]+Floor[n*u/t];
d[n_] := n + Floor[n*r/u] + Floor[n*s/u]+Floor[n*t/u];
Table[a[n], {n, 1, 120}]  (*A190508*)
Table[b[n], {n, 1, 120}]  (*A190509*)
Table[c[n], {n, 1, 120}]  (*A054770*)
Table[d[n], {n, 1, 120}]  (*A190511*)

Crossrefs

Cf. A190509, A054770, A190511 (the other three sequences in the partition of N).

Sequence in context: A201056 A352081 A257404 * A298161 A195419 A110188

Adjacent sequences: A190505 A190506 A190507 * A190509 A190510 A190511

Keyword

nonn

Author

Clark Kimberling, May 11 2011

Status

approved