A189785 a(n) = n+floor(n*r/s)+floor(nt/s); r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 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

Offset

1,1

Comments

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

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

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

c(n)=n+[nr/t]+[ns/t], where []=floor.

Taking r=Pi/2, s=arcsin(5/13), t=arcsin(12/13) gives

a=A005408, b=A189785, c=A189786. Note that r=s+t.

a(n) first differs from A017137(n-1) at n=48 (a(48)=380 but A017137(47)=382). - Nathaniel Johnston, May 16 2011

Links

Table of n, a(n) for n=1..69.

Mathematica

r=Pi/2; s=ArcSin[5/13]; t=ArcSin[12/13];
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (*A005408*)
Table[b[n], {n, 1, 120}]  (*A189785*)
Table[c[n], {n, 1, 120}]  (*A189786*)
Table[b[n]/2, {n, 1, 120}]  (*A189787*)
Table[c[n]/2, {n, 1, 120}]  (*A004773*)

Crossrefs

Cf. A189786, A189787, A004773.

Sequence in context: A062316 A079299 A043445 * A017137 A211998 A190522

Adjacent sequences: A189782 A189783 A189784 * A189786 A189787 A189788

Keyword

nonn

Author

Clark Kimberling, Apr 27 2011

Status

approved